Browse Course Material

Course info, instructors.

  • Prof. Tom Leighton
  • Dr. Marten van Dijk

Departments

  • Electrical Engineering and Computer Science
  • Mathematics

As Taught In

  • Computer Science
  • Applied Mathematics
  • Discrete Mathematics
  • Probability and Statistics

Learning Resource Types

Mathematics for computer science, lecture 1: introduction and proofs.

Description: Introduction to mathematical proofs using axioms and propositions. Covers basics of truth tables and implications, as well as some famous hypotheses and conjectures.

Speaker: Tom Leighton

  • Download video
  • Download transcript

MIT Open Learning

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Mathematics LibreTexts

3.1: An Introduction to Proof Techniques

  • Last updated
  • Save as PDF
  • Page ID 8393

  • Harris Kwong
  • State University of New York at Fredonia via OpenSUNY

A proof is a logical argument that verifies the validity of a statement. A good proof must be correct, but it also needs to be clear enough for others to understand. In the following sections, we want to show you how to write mathematical arguments. It takes practice to learn how to write mathematical proofs; you have to keep trying! We would like to start with some suggestions.

  • Write at the level of your peers . A common question asked by many students is: how much detail should I include in a proof? One simple guideline is to write at the level that your peers can understand. Although you can skip the detailed computation, be sure to include the major steps in an argument.
  • Use symbols and notations appropriately . Do not use mathematical symbols as abbreviations. For example, do not write “\(x\) is a number \(>4\).” Use “\(x\) is a number greater than 4” instead. Do not use symbols excessively either. It is often clearer if we express our idea in words. Finally, do not start a sentence with a symbol, as in “Suppose \(xy>0\). \(x\) and \(y\) have the same signs.” It would look better if we combine the two sentences, and write “Suppose \(xy>0\), then \(x\) and \(y\) have the same signs.”
  • Display long and important equations separately . Make the key mathematical results stand out by displaying them separately on their own. Be sure to center these expressions. Number them if you need to refer to them later. See Examples \(1.3.1\) and \(1.3.2\) in Section 1.3 .
  • Write in complete sentences, with proper usage of grammar and punctuation . A proof is, after all, a piece of writing. It should conform to the usual writing rules. Use complete sentences, and do not forget to check the grammar and punctuation.
  • Start with a draft . Prepare a draft. When you feel it is correct, start revising it: check the accuracy, remove redundancy, and simplify the sentence structure. Organize the argument into short paragraphs to enhance the readability of a proof. Go over the proof and refine it further.

Some proofs only require direct computation.

Example \(\PageIndex{1}\label{eg:pfintro-01}\)

Let \(a\) and \(b\) be two rational numbers such that \(a<b\). Show that the weighted average \(\frac{1}{3}\,a+\frac{2}{3}\,b\) is a rational number between \(a\) and \(b\).

Since \(a\) and \(b\) are rational numbers, we can write \(a=\frac{m}{n}\) and \(b=\frac{p}{q}\) for some integers \(m\), \(n\), \(p\), and \(q\), where \(n,q\neq0\). Then \[\frac{1}{3}\,a+\frac{2}{3}\,b = \frac{1}{3}\cdot\frac{m}{n} + \frac{2}{3}\cdot\frac{p}{q} = \frac{mq+2np}{3nq} \nonumber\] is clearly a rational number because \(mq+2np\) and \(3np\) are integers, and \(3nq\neq0\). Since \(a<b\), we know \(b-a>0\). It follows that \[\left(\frac{1}{3}\,a+\frac{2}{3}\,b\right) - a = \frac{2}{3}\,(b-a) > 0, \nonumber\] which means \(\frac{1}{3}\,a+\frac{2}{3}\,b > a\). In a similar fashion, we also find \(\frac{1}{3}\,a+\frac{2}{3}\,b < b\). Thus, \(\frac{1}{3}\,a+\frac{2}{3}\,b\) is a rational number between \(a\) and \(b\).

hands-on Exercise \(\PageIndex{1}\label{he:pfintro-01}\)

Show that \(\frac{1}{3}\,a+\frac{2}{3}\,b\) is closer to \(b\) than to \(a\).

Compute the distance between \(a\) and \(\frac{1}{3}\,a+\frac{2}{3}\,b\), and compare it to the distance between \(\frac{1}{3}\,a+\frac{2}{3}\,b\) and \(b\).

Sometimes, we can use a constructive proof when a proposition claims that certain values or quantities exist.

Example \(\PageIndex{2}\label{eg:pfintro-02}\)

Prove that every positive integer can be written in the form of \(2^e t\) for some nonnegative integer \(e\) and some odd integer \(t\).

The problem statement only says “every positive integer.” It often helps if we assign a name to the integer; it will make it easier to go through the discussion. Consequently, we customarily start a proof with the phrase “Let \(n\) be … ”

Let \(n\) be a positive integer. Keep dividing \(n\) by 2 until an odd number \(t\) remains. Let \(e\) be the number of times we factor out a copy of 2. It is clear that \(e\) is nonnegative, and we have found \(n=2^e t\).

Example \(\PageIndex{3}\label{eg:pfintro-03}\)

Given any positive integer \(n\), show that there exist \(n\) consecutive composite positive integers.

For each positive integer \(n\), we claim that the \(n\) integers \[(n+1)!+2, \quad (n+1)!+3, \quad \ldots \quad (n+1)!+n, \quad (n+1)!+(n+1) \nonumber\] are composite. Here is the reason. For each \(i\), where \(2\leq i\leq n+1\), the integer \[\begin{aligned} (n+1)!+i &=& 1\cdot2\cdot3\,\cdots(i-1)i(i+1)\cdots\,(n+1)+i \\ &=& i\,[\,1\cdot2\cdot3\,\cdots(i-1)(i+1)\cdots\,(n+1)+1\,] \end{aligned} \nonumber\] is divisible by \(i\) and greater than \(i\), and hence is composite.

hands-on Exercise \(\PageIndex{3}\label{he:pfintro-03}\)

Construct five consecutive positive integers that are composite. Verify their compositeness by means of factorization.

Example \(\PageIndex{4}\label{eg:pfintro-04}\)

Let \(m\) and \(n\) be positive integers. Show that, if \(mn\) is even, then an \(m\times n\) chessboard can be fully covered by non-overlapping dominoes.

This time, the names \(m\) and \(n\) have already been assigned to the two positive integers. Thus, we can refer to them in the proof without an introduction.

Since \(mn\) is even, one of the two integers \(m\) and \(n\) must be even. Without loss of generality (since the other case is similar), we may assume \(m\), the number of rows, is even. Then \(m=2t\) for some integer \(t\). Each column can be filled with \(m/2=t\) non-overlapping dominoes placed vertically. As a result, the entire chessboard can be covered with \(nt\) non-overlapping vertical dominoes.

hands-on Exercise \(\PageIndex{4}\label{he:pfintro-04}\)

Show that, between any two rational numbers \(a\) and \(b\), where \(a<b\), there exists another rational number.

Try the midpoint of the interval \([a,b]\).

Exercise \(\PageIndex{5}\label{he:pfintro-05}\)

Show that, between any two rational numbers \(a\) and \(b\), where \(a<b\), there exists another rational number closer to \(b\) than to \(a\).

Use a weighted average of \(a\) and \(b\).

Sometimes a non-constructive proof can be used to show the existence of a certain quantity that satisfies some conditions. We have learned two such existence theorems from calculus.

Theorem \(\PageIndex{1}\) (Mean Value Theorem)

Let \(f\) be a differentiable function defined over a closed interval \([a,b]\). Then there exists a number \(c\) strictly inside the open interval \((a,b)\) such that \(f'(c) = \frac{f(b)-f(a)}{b-a}\).

Theorem \(\PageIndex{2} \label{thm:IVT}\) (Intermediate Value Theorem)

Let \(f\) be a function that is continuous over a closed interval \([a,b]\). Then \(f\) assumes all values between \(f(a)\) and \(f(b)\). In other words, for any value \(t\) between \(f(a)\) and \(f(b)\), there exists a number \(c\) inside \([a,b]\) such that \(f(c)=t\).

Both results only guarantee the existence of a number \(c\) with some specific property; they do not tell us how to find this number \(c\). Nevertheless, the Mean-Value Theorem plays a very important role in analysis; many of its applications are beyond the scope of this course. We could, however, demonstrate an application of the Intermediate Value Theorem.

Corollary \(\PageIndex{3}\label{cor:IVT}\)

Let \(f\) be a continuous function defined over a closed interval \([a,b]\). If \(f(a)\) and \(f(b)\) have opposite signs, then the equation \(f(x)=0\) has a solution between \(a\) and \(b\).

According to the Intermediate Value Theorem, \(f(x)\) can take on any value between \(f(a)\) and \(f(b)\). Since they have opposite signs, 0 is a number between them. Hence, \(f(c)=0\) for some number \(c\) between \(a\) and \(b\).

Example \(\PageIndex{5}\label{eg:pfintro-05}\)

The function \(f(x)=5x^3-2x-1\) is a polynomial function, which is known to be continuous over the real numbers. Since \(f(0)=-1\) and \(f(1)=2\), Corollary 3.1.3 implies that there exists a number between 0 and 1 such that \(5x^3-2x-1=0\).

Summary and Review

  • Sometimes we can prove a statement by showing how the result can be obtained through a construction, and we can describe the construction in an algorithm.
  • Sometimes all we need to do is apply an existence theorem to verify the existence of a certain quantity.

Course details

Introduction to mathematical proofs.

This is an Online course which requires your attendance to weekly meetings which take place online using Microsoft Teams meetings.

This short course will combine pre-recorded lectures with live, weekly, online meetings where discussion and debate will take place between students and the tutor. Visit our How our WOW courses work page for full details.

This course will close for enrolment 7 days prior to its start date.

Is there only one way to prove a mathematical statement? Do we always follow a deductive method when trying to reach a new conclusion? The quick answer is: no. Mathematics has developed a variety of techniques over the centuries in the continuous pursuit of finding the clearest or most elegant proof for a problem. All these methods show a highly creative thinking process that exploits experience, intuition, and even a sense of adventure. 

By relating the presentation of different types of proofs to historical examples and anecdotes from the lives of famous Mathematicians, from Euclid to Andrew Wiles, the students will embark on a rich investigative journey that will show them why mathematicians can be considered to be part detectives and part artists, thus developing their analytical skills and a broader overview of what a proof is. 

Programme details

Courses start: 24 Apr 2024

Week 0: Course Orientation

Week 1: Introduction to Logic and Sets

Week 2: Proof Techniques: Direct Proof

Week 3: Proof Techniques: Mathematical Induction

Week 4: Relations and Functions

Week 5: Proof Techniques: Proof by Cases and Counterexamples

Week 6: Axiomatic Systems

Week 7: Quantifiers and Logic

Week 8: Proof Techniques: Proof by Contrapositive and Proof by Exhaustion

Week 9: Divisibility Proofs

Week 10: Proof Techniques: Proof by Contradiction

Recommended reading

All weekly class students may become borrowing members of the Rewley House Continuing Education Library for the duration of their course. Prospective students whose courses have not yet started are welcome to use the Library for reference. More information can be found on the Library website.

There is a Guide for Weekly Class students which will give you further information.

Availability of titles on the reading list (below) can be checked on SOLO , the library catalogue.

Preparatory reading

  • How to Prove It: A Structured Approach / Daniel J. Velleman
  • How to Read and Do Proofs: An Introduction to Mathematical Thought Processes / Daniel Solow
  • How to Solve It: A New Aspect of Mathematical Method / George Pólya
  • Proofs and Refutations / Imre Lakatos
  • The Art of Proof: Basic Training for Deeper Mathematics / Matthias Beck and Ross Geoghegan

Recommended Reading List

Certification

To complete the course and receive a certificate, you will be required to attend and participate in at least 80% of the live sessions on the course and pass your final assignment. Upon successful completion, you will receive a link to download a University of Oxford digital certificate. Information on how to access this digital certificate will be emailed to you after the end of the course. The certificate will show your name, the course title and the dates of the course you attended. You will be able to download your certificate or share it on social media if you choose to do so.

If you are in receipt of a UK state benefit, you are a full-time student in the UK or a student on a low income, you may be eligible for a reduction of 50% of tuition fees. Please see the below link for full details:

Concessionary fees for short courses

Dr Niccolò Salvatori

Dr Niccolo Salvatori is a Guest Teacher at the London School of Economics and a former Honorary Research Associate of King's College London. In the past, he has lectured Calculus for the University of California, Berkeley for study abroad programmes and has recently joined the Department for Continuing Education, Oxford. Niccolo also teaches at Secondary School and Sixth Form level, including Further Mathematics, and has been creating and delivering projects and enrichment courses for A-Level students since 2017. His interests are in Analysis and Geometry, but he is passionate about Logic and has great admiration for the work of Alan Turing and its far-reaching consequences.

Course aims

To introduce the basic tools of correct thinking and the logical techniques applied in mathematics to create proofs.

Course objectives:

  • To develop a solid understanding of the foundations of mathematical proofs, including logic, sets, and proof techniques.
  • To cultivate critical thinking skills and the ability to construct rigorous mathematical arguments.
  • To enhance problem-solving skills through engaging in-class discussions, problem-solving activities, and practical applications of proof techniques.
  • To understand how proofs have evolved through the centuries thanks to the work and dedication of famous mathematicians.

Teaching methods

Students will have access to a pre-recorded lecture to be watched in advance of the weekly online session.

Learning outcomes

By the end of the course, students will be expected to:

  • have a solid understanding of the basic principles of mathematical proofs, including logic, sets, and proof techniques;
  • have improved their critical thinking skills and ability to construct rigorous arguments through class discussions and problem-solving activities, appreciating how this can be applied in everyday life;
  • have knowledge of the historical development of proofs thanks to famous mathematicians' work, providing them with a broader perspective on the subject. 

Assessment methods

Short exercises and a short report.

Students must submit a completed Declaration of Authorship form at the end of term when submitting your final piece of work. CATS points cannot be awarded without the aforementioned form - Declaration of Authorship form

Application

We will close for enrolments 7 days prior to the start date to allow us to complete the course set up. We will email you at that time (7 days before the course begins) with further information and joining instructions. As always, students will want to check spam and junk folders during this period to ensure that these emails are received.

To earn credit (CATS points) for your course you will need to register and pay an additional £10 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online.

Please use the 'Book' or 'Apply' button on this page. Alternatively, please complete an  enrolment form (Word)  or  enrolment form (Pdf) .

Level and demands

GCSE Mathematics 

Essential knowledge:

• GCSE level algebra.

• GCSE level number facts and procedures (e.g.: basic operations, factorization, sequences of numbers, etc.).

Desirable knowledge

• Definition of a function and composition of functions.

Students who register for CATS points will receive a Record of CATS points on successful completion of their course assessment.

To earn credit (CATS points) you will need to register and pay an additional £10 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online.

Coursework is an integral part of all weekly classes and everyone enrolled will be expected to do coursework in order to benefit fully from the course. Only those who have registered for credit will be awarded CATS points for completing work at the required standard.

Students who do not register for CATS points during the enrolment process can either register for CATS points prior to the start of their course or retrospectively from the January 1st after the current full academic year has been completed. If you are enrolled on the Certificate of Higher Education you need to indicate this on the enrolment form but there is no additional registration fee.

Most of the Department's weekly classes have 10 or 20 CATS points assigned to them. 10 CATS points at FHEQ Level 4 usually consist of ten 2-hour sessions. 20 CATS points at FHEQ Level 4 usually consist of twenty 2-hour sessions. It is expected that, for every 2 hours of tuition you are given, you will engage in eight hours of private study.

Credit Accumulation and Transfer Scheme (CATS)

Terms & conditions for applicants and students

Information on financial support

introduction to proof assignment

  • Engineering Mathematics
  • Discrete Mathematics
  • Operating System
  • Computer Networks
  • Digital Logic and Design
  • C Programming
  • Data Structures
  • Theory of Computation
  • Compiler Design
  • Computer Org and Architecture

Related Articles

  • Discrete Mathematics Tutorial

Mathematical Logic

  • Propositional Logic
  • Discrete Mathematics - Applications of Propositional Logic
  • Propositional Equivalences
  • Predicates and Quantifiers
  • Mathematics | Some theorems on Nested Quantifiers
  • Rules of Inference

Mathematics | Introduction to Proofs

Sets and relations.

  • Set Theory - Definition, Types of Sets, Symbols & Examples
  • Types Of Sets
  • Irreflexive Relation on a Set
  • Reflexive Relation on Set
  • Transitive Relation on a Set
  • Set Operations
  • Types of Functions
  • Mathematics | Sequence, Series and Summations
  • Representation of Relation in Graphs and Matrices
  • What are Relations?
  • Closure of Relations

Mathematical Induction

  • Pigeonhole Principle
  • Mathematics | Generalized PnC Set 1
  • Discrete Maths | Generating Functions-Introduction and Prerequisites
  • Inclusion Exclusion principle and programming applications

Boolean Algebra

  • Properties of Boolean Algebra
  • Number of Boolean functions
  • Minimization of Boolean Functions
  • Linear Programming

Ordered Sets & Lattices

  • Elements of POSET
  • Partial Order Relation on a Set
  • Axiomatic Approach to Probability
  • Properties of Probability

Probability Theory

  • Mathematics | Probability Distributions Set 1 (Uniform Distribution)
  • Mathematics | Probability Distributions Set 2 (Exponential Distribution)
  • Mathematics | Probability Distributions Set 3 (Normal Distribution)
  • Mathematics | Probability Distributions Set 5 (Poisson Distribution)

Graph Theory

  • Graph and its representations
  • Mathematics | Graph Theory Basics - Set 1
  • Types of Graphs with Examples
  • Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph
  • How to find Shortest Paths from Source to all Vertices using Dijkstra's Algorithm
  • Prim’s Algorithm for Minimum Spanning Tree (MST)
  • Kruskal’s Minimum Spanning Tree (MST) Algorithm
  • Check whether a given graph is Bipartite or not
  • Eulerian path and circuit for undirected graph

Special Graph

  • Introduction to Graph Coloring
  • Edge Coloring of a Graph
  • Check if a graph is Strongly, Unilaterally or Weakly connected
  • Biconnected Components
  • Strongly Connected Components

Group Theory

  • Homomorphism & Isomorphism of Group
  • Group Isomorphisms and Automorphisms
  • Algebraic Structure

GATE PYQs and LMN

  • Last Minute Notes – Discrete Mathematics
  • Discrete Mathematics - GATE CSE Previous Year Questions

Discrete Mathematics MCQ

  • Propositional and First Order Logic.
  • Set Theory & Algebra
  • Combinatorics
  • Top MCQs on Graph Theory in Mathematics
  • Numerical Methods and Calculus

Mathematical proof is an argument we give logically to validate a mathematical statement. In order to validate a statement, we consider two things: A statement and Logical operators .  A statement is either true or false but not both. Logical operators are AND, OR, NOT, If then, and If and only if. Coupled with quantifiers like for all and there exists. We apply operators on the statement to check the correctness of it.  Types of mathematical proofs:  

  • Proof by cases –   In this method, we evaluate every case of the statement to conclude its truthiness.  Example: For every integer x, the integer x(x + 1) is even  Proof: If x is even, hence, x = 2k for some number k. now the statement becomes:   

which is divisible by 2, hence it is even.  If x is odd, hence x = 2k + 1 for some number k, now the statement becomes: 

which is again divisible by 2 and hence in both cases we proved that x(x+1) is even. 

  • Proof by contradiction –   We assume the negation of the given statement and then proceed to conclude the proof.  Example: Prove that sqrt(2) is irrational  Suppose sqrt(2) is rational. 

for some integers a and b with b != 0.  Let us choose integers a and b with sqrt(2) = a/b, such that b is positive and as small as possible. (Well-Ordering Principle) 

Since a^2 is even, it follows that a is even.  a = 2k for some integer k, so a^2 = 4k^2  b^2 = 2k^2. Since b^2 is even, it follows that b is even.  Since a and b are both even, a/2 and b/2 are integers with b/2 > 0, and sqrt(2) = (a/2)/(b/2), because (a/2)/(b/2) = a/b.  But it contradicts our assumption b is as small as possible. Therefore sqrt(2) cannot be rational. 

  • Proof by induction –   The Principle of Mathematical Induction (PMI). Let P(n) be a statement about the positive integer n. If the following are true: 

Example: For every positive integer n, 

Proof:   Base case: If n = 1, 

  • Inductive step:   Suppose that for a given n there exists Z+, 

Our goal is to show that: 

Add n + 1 both sides to equation (i), we get, 

  • Direct Proof –   when we want to prove a conditional statement p implies q, we assume that p is true, and follow implications to get to show that q is then true.  It is Mostly an application of hypothetical syllogism, [(p → r) ∧ (r → q)] → (p → q)]  We just have to find the propositions that lead us to q.  Theorem: If m is even and n is odd, then their sum is odd  Proof:   Since m is even, there is an integer j such that m = 2j.  Since n is odd, there is an integer k such that n = 2k+1. Then,   

Since j+k is an integer, we see that m+n is odd. 

Please Login to comment...

  • anikaseth98

Improve your Coding Skills with Practice

 alt=

What kind of Experience do you want to share?

18.S097 Special Subject in Mathematics: Introduction to Proofs

Logo for Milne Publishing

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Part I: Propositional Logic

4.1  A problem with semantic demonstrations of validity

Given that we can test an argument for validity, it might seem that we have a fully developed system to study arguments.  However, there is a significant practical difficulty with our semantic method of checking arguments using truth tables (you may have already noted what this practical difficulty is, when you did problems 1e and 2e of chapter 3).  Consider the following argument:

Alison will go to the party.

If Alison will go to the party, then Beatrice will.

If Beatrice will go to the party, then Cathy will.

If Cathy will go to the party, then Diane will.

If Diane will go to the party, then Elizabeth will.

If Elizabeth will go to the party, then Fran will.

If Fran will go to the party, then Giada will.

If Giada will go to the party, then Hilary will.

If Hillary will go to the party, then Io will.

If Io will go to the party, then Julie will.

Julie will go to the party.

Most of us will agree that this argument is valid.  It has a rather simple form, in which one sentence is related to the previous sentence, so that we can see the conclusion follows from the premises.  Without bothering to make a translation key, we can see the argument has the following form.

However, if we are going to check this argument, then the truth table will require 1024 rows!  This follows directly from our observation that for arguments or sentences composed of n atomic sentences, the truth table will require 2 n  rows.  This argument contains 10 atomic sentences.  A truth table checking its validity must have 2 10  rows, and 2 10 =1024.  Furthermore, it would be trivial to extend the argument for another, say, ten steps, but then the truth table that we make would require more than a million rows!

For this reason, and for several others (which become evident later, when we consider more advanced logic), it is very valuable to develop a syntactic proof method.  That is, a way to check proofs not using a truth table, but rather using rules of syntax.

Here is the idea that we will pursue.  A valid argument is an argument such that, necessarily, if the premises are true, then the conclusion is true.  We will start just with our premises.  We will set aside the conclusion, only to remember it as a goal.  Then, we will aim to find a reliable way to introduce another sentence into the argument, with the special property that, if the premises are true, then this single additional sentence to the argument must also be true.  If we could find a method to do that, and if after repeated applications of this method we were able to write down our conclusion, then we would know that, necessarily, if our premises are true then the conclusion is true.

The idea is more clear when we demonstrate it.  The method for introducing new sentences will be called “inference rules”.  We introduce our first inference rules for the conditional.  Remember the truth table for the conditional:

Look at this for a moment.  If we have a conditional like (P→Q)  (looking at the truth table above, remember that this would meant that we let Φ  be P  and Ψ  be Q ), do we know whether any other sentence is true?  From (P →Q)  alone we do not.  Even if (P→Q)  is true, P  could be false or Q  could be false.  But what if we have some additional information?  Suppose we have as premises both (P →Q)  and P .  Then, we would know that if those premises were true, Q  must be true.  We have already checked this with a truth table.

The first row of the truth table is the only row where all of the premises are true; and for it, we find that Q  is true.  This, of course, generalizes to any conditional.  That is, we have that:

We now capture this insight not using a truth table, but by introducing a rule.  The rule we will write out like this:

This is a syntactic rule.  It is saying that whenever we have written down a formula in our language that has the shape of the first row (that is, whenever we have a conditional), and whenever we also have written down a formula that has the shape in the second row (that is, whenever we also have written down the antecedent of the conditional), then go ahead, whenever you like, and write down a formula like that in the third row (the consequent of the conditional).  The rule talks about the shape of the formulas, not their meaning.  But of course we justified the rule by looking at the meanings.

We describe this by saying that the third line is “derived” from the earlier two lines using the inference rule.

This inference rule is old.  We are, therefore, stuck with its well-established, but not very enlightening, name:  “modus ponens”.  Thus, we say, for the above example, that the third line is derived from the earlier two lines using modus ponens.

4.2  Direct proof

We need one more concept:  that of a proof.  Specifically, we’ll start with the most fundamental kind of proof, which is called a “direct proof”.  The idea of a direct proof is:  we write down as numbered lines the premises of our argument.  Then, after this, we can write down any line that is justified by an application of an inference rule to earlier lines in the proof.  When we write down our conclusion, we are done.

Let us make a proof of the simple argument above, which has premises (P→Q)  and P , and conclusion Q .  We start by writing down the premises and numbering them.   There is a useful bit of notation that we can introduce at this point.  It is known as a “Fitch bar”, named after a logician Frederic Fitch, who developed this technique.  We will write a vertical bar to the left, with a horizontal line indicating that the premises are above the line.

\[ \fitchprf{\pline[1.]{(P \lif Q)}\\ \pline[2.]{P}} { } \]

It is also helpful to identify where these steps came from.  We can do that with a little explanation written out to the right.

\[ \fitchprf{\pline[1.] {(P \lif Q)} [premise]\\ \pline[2.]{P} [premise] } { } \]

Now, we are allowed to write down any line that follows from an earlier line using an inference rule.

\[ \fitchprf{\pline[1.] {(P \lif Q)} [premise]\\ \pline[2.]{P} [premise] } { \pline[3.]{Q} } \]

And, finally, we want a reader to understand what rule we used, so we add that into our explanation, identifying the rule and the lines used.

\[ \fitchprf{\pline[1.] {(P \lif Q)} [premise]\\ \pline[2.]{P} [premise] } { \pline[3.]{Q} [modus ponens, 1, 2] } \]

That is a complete direct proof.

Notice a few things.  The numbering of each line, and the explanations to the right, are bookkeeping; they are not part of our argument, but rather are used to explain our argument.  However, always do them because, it is hard to understand a proof without them.  Also, note that our idea is that the inference rule can be applied to any earlier line, including lines themselves derived using inference rules.  It is not just premises to which we can apply an inference rule.  Finally, note that we have established that this argument must be valid.  From the premises, and an inference rule that preserves validity, we have arrived at the conclusion.  Necessarily, the conclusion is true, if the premises are true.

The long argument that we started the chapter with can now be given a direct proof.

\[ \fitchprf{\pline[1.] {P} [premise]\\ \pline[2.]{(P \lif Q)} [premise]\\ \pline[3.]{(Q \lif R)} [premise]\\ \pline[4.]{(R \lif S)} [premise]\\ \pline[5.]{(S \lif T)} [premise]\\ \pline[6.]{(T \lif U)} [premise]\\ \pline[7.]{(U \lif V)} [premise]\\ \pline[8.]{(V \lif W)} [premise]\\ \pline[9.]{(W \lif X)} [premise]\\ \pline[10.]{(X \lif Y)} [premise]\\ } { \pline[11.]{Q} [modus ponens, 2, 1]\\ \pline[12.]{R} [modus ponens, 3, 11]\\ \pline[13.]{S} [modus ponens, 4, 12]\\ \pline[14.]{T} [modus ponens, 5, 13]\\ \pline[15.]{U} [modus ponens, 6, 14]\\ \pline[16.]{V} [modus ponens, 7, 15]\\ \pline[17.]{W} [modus ponens, 8, 16]\\ \pline[18.]{X} [modus ponens, 9, 17]\\ \pline[19.]{Y} [modus ponens, 10, 18] } \]

From repeated applications of modus ponens, we arrived at the conclusion.  If lines 1 through 10 are true, line 19 must be true.  The argument is valid.  And, we completed it with 19 steps, as opposed to writing out 1024 rows of a truth table.

We can see now one of the very important features of understanding the difference between syntax and semantics.  Our goal is to make the syntax of our language perfectly mirror its semantics.  By manipulating symbols, we manage to say something about the world.  This is a strange fact, one that underlies one of the deeper possibilities of language, and also, ultimately, of computers.

4.3  Other inference rules

We can now introduce other inference rules.  Looking at the truth table for the conditional again, what else do we observe?  Many have noted that if the consequent of a conditional is false, and the conditional is true, then the antecedent of the conditional must be false.  Written out as a semantic check on arguments, this will be:

(Remember how we have filled out the truth table.  We referred to those truth tables used to define “→” and “ ¬” , and then for each row of this table above, we filled out the values in each column based on that definition.)

What we observe from this truth table is that when both (Φ→Ψ)  and ¬Ψ  are true, then ¬Φ  is true.  Namely, this can be seen in the last row of the truth table.

This rule, like the last, is old, and has a well-established name:  “modus tollens”.  We represent it schematically with

What about negation?  If we know a sentence is false, then this fact alone does not tell us about any other sentence.  But what if we consider a negated negation sentence?  Such a sentence has the following truth table.

We can introduce a rule that takes advantage of this observation.  In fact, it is traditional to introduce two rules, and lump them together under a common name.  The rules’ name is “double negation”.  Basically, the rule says we can add or take away two negations any time.  Here are the two schemas for the two rules:

Finally, it is sometimes helpful to be able to repeat a line.  Technically, this is an unnecessary rule, but if a proof gets long, we often find it easier to understand the proof if we write a line over again later when we find we need it again.  So we introduce the rule “repeat”.

4.4  An example

Here is an example that will make use of all three rules.  Consider the following argument:

We want to check this argument to see if it is valid.

To do a direct proof, we number the premises so that we can refer to them when using inference rules.

\[ \fitchprf{\pline[1.] {(Q \lif P)} [premise]\\ \pline[2.]{(\lnot Q \lif R)} [premise]\\ \pline[3.]{\lnot R} [premise]\\ } { } \]

And, now, we apply our inference rules.  Sometimes, it can be hard to see how to complete a proof.  In the worst case, where you are uncertain of how to proceed, you can apply all the rules that you see are applicable and then, assess if you have gotten closer to the conclusion; and repeat this process.  Here in any case is a direct proof of the sought conclusion.

\[ \fitchprf{\pline[1.] {(Q \lif P)} [premise]\\ \pline[2.]{(\lnot Q \lif R)} [premise]\\ \pline[3.]{\lnot R} [premise]\\ } { \pline[4.]{\lnot \lnot Q}[modus tollens, 2, 3]\\ \pline[5.]{Q}[double negation, 4]\\ \pline[6.]{P}[modus ponens, 1, 5]\\ } \]

Developing skill at completing proofs merely requires practice.  You should strive to do as many problems as you can.

4.5  Problems

1. Complete a direct derivation (also called a “direct proof”) for each of the following arguments, showing that it is valid. You will need the rules modus ponens, modus tollens, and double negation.

  • Premises: (P→Q) , ¬¬P . Show: ¬¬Q .
  • Premises: Q , (¬P→¬Q) . Show: P .
  • Premises: ¬Q , (¬Q→S) . Show: S .
  • Premises: ¬S , (¬Q→S) . Show: Q .
  • Premises: (S→¬Q) , (P→S) , ¬¬P . Show: ¬Q .
  • Premises: (T→P) , (Q→S) , (S→T) , ¬P . Show: ¬Q .
  • Premises: R, P, (P →  (R →  Q)) . Show: Q .
  • Premises: ((R→S)→Q) , ¬Q , (¬(R→S)→V) . Show: V .
  • Premises:   (P→(Q→R)) ,  ¬(Q→R) .  Show:   ¬P .
  • Premises:   (¬(Q→R)→P) , ¬P , Q . Show:   R .
  • Premises:   P , (P→R) , (P→(R→Q)) .  Show:   Q .
  • Premises:   ¬R , (S→R) , P , (P→(T→S)) .  Show:   ¬T .
  • Premises:   P , (P→Q) , (P→R) , (Q→(R→S)) .  Show:   S .
  • Premises:   (P→(Q→R)) , P , ((Q→R)→¬S)) , ((T→V)→S) .  Show:   ¬(T→V) .

2. In normal colloquial English, write your own valid argument with at least two premises. Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like logic).  Translate it into propositional logic and use a direct proof to show it is valid.

3. In normal colloquial English, write your own valid argument with at least three premises. Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like logic).  Translate it into propositional logic and use a direct proof to show it is valid.

4. Make your own key to translate into propositional logic the portions of the following argument that are in bold.  Using a direct proof, prove that the resulting argument is valid.

Inspector Tarski told his assistant, Mr. Carroll, “ If Wittgenstein had mud on his boots, then he was in the field .  Furthermore, if Wittgenstein was in the field, then he is the prime suspect for the murder of Dodgson.  Wittgenstein did have mud on his boots.   We conclude, Wittgenstein is the prime suspect for the murder of Dodgson. ”

A Concise Introduction to Logic Copyright © 2017 by Craig DeLancey is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

IMAGES

  1. PPT

    introduction to proof assignment

  2. How to Write an Introduction for an Assignment

    introduction to proof assignment

  3. How To Write An Assignment Introduction With Example by John Wright

    introduction to proof assignment

  4. Introduction to Proof Assignment and Quiz Flashcards _ Quizlet.pdf

    introduction to proof assignment

  5. How to Write An Assignment Introduction Like A Pro

    introduction to proof assignment

  6. 1.3.2B Introduction to Proof

    introduction to proof assignment

VIDEO

  1. Character Design

  2. Proof of work

  3. 1. Introduction

  4. NPTEL Introduction to Operations Research ASSIGNMENT ANSWERS WEEK-3

  5. PROOFS

  6. QCAA Specialist Mathematics: Introduction to proof by Induction (Part 2)

COMMENTS

  1. Introduction to Proof Assignment and Quiz Flashcards

    A: given. B: measure of angle ABC = 90. C: angle addition postulate. D: 2 times the measure of angle CBD = 90. Given: m∠A + m∠B = m∠B + m∠C. Prove: m∠C = m∠A. Write a paragraph proof to prove the statement. We are given that the sum of the measures of angles A and B is equal to the sum of the measures of angles B and C.

  2. Introduction to Proof Flashcards

    reflexive. A two-column proof. b. contains a table with a logical series of statements and reasons that reach a conclusion. A paragraph proof. d. contains a set of sentence explaining the steps needed to reach a conclusion. Proof: We are given that m<AEB = 45° and <AEC is a right angle. The measure of <AEC is 90° by the definition of a right ...

  3. Introduction to Proof Flashcards

    Check all that apply. In a paragraph proof, statements and their justifications are written in sentences in a logical order. A two-column proof consists of a list statements and the reasons the statements are true. A paragraph proof is a two-column proof in sentence form. A flowchart proof includes a logical series of statements in boxes with ...

  4. PDF Mathematical Proofs

    Our First Proof! 😃 Theorem: If n is an even integer, then n2 is even. Proof:Let n be an even integer. Since n is even, there is some integer k such that n = 2k. This means that n2 = (2k)2 = 4k2 = 2(2k2). From this, we see that there is an integer m (namely, 2k2) where n2 = 2m. Therefore, n2 is even. To prove a statement of the form "If P ...

  5. Math 201

    Basic Analysis I Lebl ISBN: 9781718862401 Syllabus Sets, logic, proofs in algebra and calculus. Objectives This class is designed to guide students through the transition from example-based calculus courses to advanced proof-based classes in mathematics.

  6. Lecture 1: Introduction and Proofs

    Video Lectures Lecture 1: Introduction and Proofs Description: Introduction to mathematical proofs using axioms and propositions. Covers basics of truth tables and implications, as well as some famous hypotheses and conjectures. Speaker: Tom Leighton Transcript Download video Download transcript

  7. PDF 18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015)

    1. Introduction The goal for this course is to provide a quick, and hopefully somewhat gentle, introduction to the task of formulating and writing mathematical proofs. We begin by discussing some basic ideas of logic and sets which form the basic ingredients in our mathematical language, and conclude our discussion for the day with a few examples.

  8. PDF AnIntroductiontoProofTheory

    Introduction to Proof Theory 3 The study of Proof Theory is traditionally motivated by the problem of formaliz-ing mathematical proofs; the original formulation of flrst-order logic by Frege [1879] ... A truth assignment consists of an assignment of True/False values to the propositional variables, i.e., a truth assignment is a mapping ...

  9. 3.1: An Introduction to Proof Techniques

    Corollary 3.1.3. Let f be a continuous function defined over a closed interval [a, b]. If f(a) and f(b) have opposite signs, then the equation f(x) = 0 has a solution between a and b. Proof. Example 3.1.5. The function f(x) = 5x3 − 2x − 1 is a polynomial function, which is known to be continuous over the real numbers.

  10. Introduction to Mathematical Proofs

    Week 0: Course Orientation. Week 1: Introduction to Logic and Sets. Week 2: Proof Techniques: Direct Proof. Week 3: Proof Techniques: Mathematical Induction. Week 4: Relations and Functions. Week 5: Proof Techniques: Proof by Cases and Counterexamples. Week 6: Axiomatic Systems. Week 7: Quantifiers and Logic. Week 8: Proof Techniques: Proof by ...

  11. PDF Warm-Up Introduction to Proof

    Introduction to Proof Deductive reasoning is the process of utilizing facts, properties, definitions, and to form a logical argument. • Solving an equation using properties to steps • that two segments are congruent by using information given in a diagram Inductive reasoning is the logical process of making generalizations

  12. Mathematics

    Courses Mathematical proof is an argument we give logically to validate a mathematical statement. In order to validate a statement, we consider two things: A statement and Logical operators . A statement is either true or false but not both. Logical operators are AND, OR, NOT, If then, and If and only if.

  13. Introduction to Proofs

    IAP 2015. Syllabus. Office: Room E18-308. Office Hours: by appointment. An introduction to writing mathematical proofs, including discussion of mathematical notation, methods of proof, and strategies for formulating and communicating mathematical arguments. Topics include: introduction to logic and sets, rational numbers and proofs of ...

  14. PDF A Brief Introduction to Proofs

    assignment, then he will fail." This does not mean a student is guaranteed to pass if he turns in an assignment. On the contrary, if he submits a poor assignment, he will likely also fail. When proving the truth of an implication, you need only consider the case when the hypothesis is true. If the hypothesis is false, then the statement is ...

  15. PDF Introduction to Proofs and Proof Strategies

    Introduction to Proofs and Proof Strategies Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof. The material revolves around possible strategies to approaching a problem without classi-fying "types of proof" or providing proof templates.

  16. Math 201: Introduction to proofs

    Learning objectives: This class is designed to guide students through the transition from example-based calculus courses to advanced proof-based classes in mathematics. Emphasis is placed on learning how to recognize and handle valid mathematical statements, to create proofs of true statements, and to disprove false statements.

  17. Introduction

    This subject offers an interactive introduction to discrete mathematics oriented toward computer science and engineering.

  18. PDF Introduction to Proof Theory

    with 0). We can represent this assignment as a function αof type Var p →{0,1} Given an assignment for propositional variables, we can inductively derive meaningful assignments for formulas as follows. Given assignments for propositional variables A and B we derive an assignment for A ∧B according to the following table. A B A ∧B 1 1 1 1 ...

  19. GCA

    RIGHT In a paragraph proof, ... Linear Pairs and Vertical Angles Assignment and Quiz. 22 terms. Ava4111. Preview. Introduction to Proof. 7 terms. andrewwww72. Preview. I WILL GET 19 SEC ON TRIG TABLE. 21 terms. Anna_Feng22. Preview. Identités remarquables. 5 terms. Lydie166.

  20. Introduction to Logic and Proofs

    Introduction to Logic and Proofs. Grade 7+. CTY-Level. Session-Based. Mathematics. Explore advanced mathematical concepts in fun and interesting ways and build a strong foundation for high school, computer science, and college-level logic coursework in this introductory logic class. Through logic puzzles, adaptive world arguments, peer ...

  21. 4. Proofs

    4. Make your own key to translate into propositional logic the portions of the following argument that are in bold. Using a direct proof, prove that the resulting argument is valid. Inspector Tarski told his assistant, Mr. Carroll, " If Wittgenstein had mud on his boots, then he was in the field.

  22. 2.2 Intro to Proofs

    Section 2.2 Intro to Proofs. G.6: Proof and Reasoning. Students apply geometric skills to making conjectures, using axioms and theorems, understanding the converse and contrapositive of a statement, constructing logical arguments, and writing geometric proofs. G.1.1: Demonstrate understanding by identifying and giving examples of undefined ...

  23. Introduction to Proof Flashcards

    a proof that lists numbered statements on the left, and corresponding numbered reasons for justification on the right. Flowchart Proof. A type of proof that uses a graphical representation. Statements are placed in boxes, and the justification for each statement is written under the box. Arrows indicate the logical flow of the statements.