Unit 1: Logic and Proofs [10 LHs]

Logic provides the formal rules used to determine whether statements and arguments are mathematically valid. This unit begins with propositional logic and logical operators, then develops logical equivalences, predicates, quantifiers, rules of inference, valid arguments, and common fallacies. It then introduces the principal techniques used to prove mathematical theorems, including direct, indirect, contradiction, equivalence, cases, existence, uniqueness, and counterexample methods. Particular attention is given to translating ordinary English into precise logical notation and to recognizing invalid reasoning.

Tested Topics

1. Predicate and Quantifiers — 3x (Nov 2024, Mar 2024, 2023)
2. Proof by Contradiction — 3x (Nov 2024, Mar 2024, 2023)
3. Propositional logic — 2x (Nov 2024, Mar 2024)
4. Translating English sentences using quantifiers — 2x (Nov 2024, Mar 2024)
5. Fallacies — 1x (2023)
6. Rules of inferences for quantified statements — 1x (2023)

Propositional Logic

Definition

Logic is the systematic study of valid reasoning. It provides rules for deciding whether conclusions follow correctly from given assumptions or premises.

Propositional logic is the branch of logic that studies propositions and compound propositions formed using logical operators.

Proposition

A proposition is a declarative sentence that has exactly one truth value: either true or false.

Examples of propositions are:

• \(2+3=5\) — true.
• \(7\) is an even integer — false.
• Kathmandu is in Nepal — true.

The following are not propositions:

• “What is your name?” — a question has no truth value.
• “Close the door.” — a command has no truth value.
• “\(x+2=7\)” — without specifying \(x\), its truth value is undetermined.

Simple and Compound Propositions

A simple proposition contains a single statement. A compound proposition is formed by joining or modifying propositions using logical operators.

For example, let

\[ p:\text{ It is raining}, \qquad q:\text{ The road is wet}. \]

Then

\[ p\rightarrow q \]

means “If it is raining, then the road is wet.”

Truth Values

The truth values are normally written as:

• \(T\) for true,
• \(F\) for false.

A truth table lists the truth value of a compound proposition for every possible combination of truth values of its component propositions.

Example

Let

\[ p:\text{ 5 is prime}, \qquad q:\text{ 5 is even}. \]

Then \(p\) is true and \(q\) is false. Therefore the statement “5 is prime and 5 is even,” represented by \(p\land q\), is false.

Logical Operators

Definition

Logical operators, also called logical connectives, combine or modify propositions to form compound propositions. The main operators are NOT, AND, OR, implication, and biconditional.

Negation: NOT

The negation of a proposition \(p\), written

\[ \neg p, \]

means “not \(p\).” It reverses the truth value of \(p\).

Conjunction: AND

The conjunction

\[ p\land q \]

means “\(p\) and \(q\).” It is true only when both propositions are true.

Disjunction: OR

The disjunction

\[ p\lor q \]

means “\(p\) or \(q\).” In standard mathematical logic, OR is inclusive: it is true when \(p\) is true, \(q\) is true, or both are true.

Implication

The conditional statement

\[ p\rightarrow q \]

means “if \(p\), then \(q\).” Here \(p\) is the hypothesis or antecedent and \(q\) is the conclusion or consequent.

An implication is false only when the hypothesis is true and the conclusion is false.

Biconditional

The biconditional

\[ p\leftrightarrow q \]

means “\(p\) if and only if \(q\).” It is true when \(p\) and \(q\) have the same truth value.

Complete Truth Table

Variants of Implication

Starting with

\[ p\rightarrow q, \]

we obtain three related statements.

The original implication and its contrapositive are logically equivalent:

\[ p\rightarrow q\equiv\neg q\rightarrow\neg p. \]

The converse and inverse are also logically equivalent:

\[ q\rightarrow p\equiv\neg p\rightarrow\neg q. \]

Worked Example: Converse, Inverse and Contrapositive

Statement: “If an integer is divisible by \(4\), then it is even.”

Let

\[ p:\text{ The integer is divisible by 4}, \]

\[ q:\text{ The integer is even}. \]

Then:

• Original: If an integer is divisible by \(4\), then it is even.
• Converse: If an integer is even, then it is divisible by \(4\). This is false; \(6\) is a counterexample.
• Inverse: If an integer is not divisible by \(4\), then it is not even. This is false; \(6\) is again a counterexample.
• Contrapositive: If an integer is not even, then it is not divisible by \(4\). This is true.

Worked Example: Truth Value of a Compound Statement

Problem: Let \(p=T\), \(q=F\), and \(r=T\). Find the truth value of

\[ (p\land\neg q)\rightarrow(r\lor q). \]

Solution:

1. \(q=F\), so \(\neg q=T\).
2. \(p=T\), so \(p\land\neg q=T\land T=T\).
3. \(r=T\), so \(r\lor q=T\lor F=T\).
4. The compound statement therefore becomes \(T\rightarrow T\), which is true.

Result:

\[ (p\land\neg q)\rightarrow(r\lor q)=T. \]

Laws of Logical Equivalences

Definition

Two compound propositions \(P\) and \(Q\) are logically equivalent if they have the same truth value for every possible assignment of truth values to their component propositions. This is written

\[ P\equiv Q. \]

Important Laws

Worked Example: Proof by a Chain of Named Laws

Problem: Prove

\[ (p\rightarrow q)\land(p\rightarrow r) \equiv p\rightarrow(q\land r). \]

Solution:

\[ \begin{aligned} (p\rightarrow q)\land(p\rightarrow r) &\equiv(\neg p\lor q)\land(\neg p\lor r) &&\text{Implication law}\\ &\equiv\neg p\lor(q\land r) &&\text{Distributive law}\\ &\equiv p\rightarrow(q\land r) &&\text{Implication law}. \end{aligned} \]

Therefore,

\[ (p\rightarrow q)\land(p\rightarrow r) \equiv p\rightarrow(q\land r). \]

Example: Negating an Implication

\[ \begin{aligned} \neg(p\rightarrow q) &\equiv\neg(\neg p\lor q)\\ &\equiv\neg\neg p\land\neg q\\ &\equiv p\land\neg q. \end{aligned} \]

Thus,

\[ \boxed{\neg(p\rightarrow q)\equiv p\land\neg q}. \]

This matches the meaning of implication: saying “it is not true that \(p\) implies \(q\)” means that \(p\) actually occurs while \(q\) fails.

Translating English Sentences

Definition

Translating English sentences into propositional logic means replacing simple statements with propositional variables and representing their logical relationships using symbols such as \(\neg\), \(\land\), \(\lor\), \(\rightarrow\), and \(\leftrightarrow\).

Common English Patterns

Worked Example

Problem: Translate the following statement:

“If I study and attend class, then I will pass the course.”

Let

\[ p:\text{ I study}, \]

\[ q:\text{ I attend class}, \]

\[ r:\text{ I pass the course}. \]

The condition contains both studying and attending class, so the hypothesis is \(p\land q\). Therefore,

\[ \boxed{(p\land q)\rightarrow r}. \]

Worked Example with Multiple Connectives

Statement: “If the server is online and the network is working, then users can connect; otherwise users cannot connect.”

Let

\[ p:\text{ The server is online}, \]

\[ q:\text{ The network is working}, \]

\[ r:\text{ Users can connect}. \]

The first part is

\[ (p\land q)\rightarrow r. \]

If “otherwise” is intended to mean that failure of the combined condition prevents connection, the second part is

\[ \neg(p\land q)\rightarrow\neg r. \]

The complete formalization is therefore

\[ [(p\land q)\rightarrow r]\land[\neg(p\land q)\rightarrow\neg r]. \]

Predicate and Quantifiers

Predicate

A predicate is a statement containing one or more variables whose truth value depends on the values assigned to those variables.

For example,

\[ P(x):x>5 \]

is not by itself a proposition because its truth value depends on \(x\). If \(x=8\), \(P(8)\) is true. If \(x=2\), \(P(2)\) is false.

Domain

The domain, or universe of discourse, specifies the values that a variable may take.

For example, the statement

\[ x^2\geq0 \]

has different interpretation depending on whether \(x\) ranges over real numbers, complex numbers, or some other domain.

Why Quantifiers Are Needed

A predicate such as \(P(x)\) normally contains a free variable and therefore does not have a definite truth value until the variable is assigned or bound. Quantifiers bind variables and describe how widely the predicate is intended to apply.

For example,

\[ P(x):x^2\geq0 \]

becomes the definite proposition

\[ \forall x\in\mathbb{R},\;x^2\geq0. \]

Universal Quantifier

The universal quantifier

\[ \forall \]

means “for all” or “for every.”

The statement

\[ \forall x\,P(x) \]

means that \(P(x)\) is true for every \(x\) in the specified domain.

Existential Quantifier

The existential quantifier

\[ \exists \]

means “there exists.”

The statement

\[ \exists x\,P(x) \]

means that at least one element of the domain makes \(P(x)\) true.

Universal and Existential Examples

Over the integers:

\[ \forall x\;(x+0=x) \]

is true.

The statement

\[ \exists x\;(x^2=9) \]

is also true because \(x=3\) and \(x=-3\) satisfy the predicate.

Negation of Quantified Statements

The fundamental rules are

\[ \neg\forall x\,P(x) \equiv \exists x\,\neg P(x), \]

and

\[ \neg\exists x\,P(x) \equiv \forall x\,\neg P(x). \]

In words:

• “Not everyone has property \(P\)” means “someone does not have property \(P\).”
• “There is nobody with property \(P\)” means “everyone lacks property \(P\).”

Example of Negating a Universal Statement

Consider

\[ \forall x\in\mathbb{R},\;x^2\geq1. \]

Its negation is

\[ \exists x\in\mathbb{R},\;x^2<1. \]

The negation is true; for example, \(x=0\).

Free and Bound Variables

A variable is bound if it lies within the scope of a quantifier. Otherwise it is free.

For example, in

\[ \forall x\;P(x,y), \]

\(x\) is bound by \(\forall x\), but \(y\) remains free.

Nested and Order in Quantifiers

Definition

A statement contains nested quantifiers when one quantifier occurs within the scope of another. The order of different quantifiers can change the meaning and truth value of a statement.

Two Important Forms

Consider a predicate \(P(x,y)\). The statement

\[ \forall x\exists y\,P(x,y) \]

means:

“For every \(x\), there is some \(y\), possibly depending on \(x\), such that \(P(x,y)\) is true.”

In contrast,

\[ \exists y\forall x\,P(x,y) \]

means:

“There is one particular \(y\) that works for every \(x\).”

Worked Example: Order Changes the Truth Value

Let the domain be the integers \(\mathbb{Z}\), and consider the predicate

\[ P(x,y):y>x. \]

Statement 1

\[ \forall x\in\mathbb{Z}\;\exists y\in\mathbb{Z}\;(y>x). \]

This is true. Given any integer \(x\), choose

\[ y=x+1. \]

Then \(y>x\).

Statement 2

\[ \exists y\in\mathbb{Z}\;\forall x\in\mathbb{Z}\;(y>x). \]

This says that there is one integer larger than every integer. It is false.

For any proposed \(y\), choose

\[ x=y+1. \]

Then \(x>y\), contradicting \(y>x\).

Therefore,

\[ \forall x\exists y\;(y>x) \]

is true over \(\mathbb{Z}\), while

\[ \exists y\forall x\;(y>x) \]

is false.

When Quantifier Order May Be Interchanged

Quantifiers of the same type can usually be reordered:

\[ \forall x\forall y\,P(x,y) \equiv \forall y\forall x\,P(x,y), \]

and

\[ \exists x\exists y\,P(x,y) \equiv \exists y\exists x\,P(x,y). \]

However, a universal and an existential quantifier generally cannot be exchanged.

Translating English Sentences Using Quantifiers

Definition

English statements involving words such as “all,” “every,” “some,” “there exists,” and “none” can be expressed precisely using predicates and quantifiers.

General Translation Patterns

Worked Example: Not All Heroes Are Brave

Let the domain be people and define

\[ H(x):x\text{ is a hero}, \]

\[ B(x):x\text{ is brave}. \]

“All heroes are brave” would be

\[ \forall x(H(x)\rightarrow B(x)). \]

Therefore “Not all heroes are brave” is

\[ \neg\forall x(H(x)\rightarrow B(x)). \]

Using quantifier negation, this is equivalent to

\[ \exists x\neg(H(x)\rightarrow B(x)). \]

Because

\[ \neg(H(x)\rightarrow B(x)) \equiv H(x)\land\neg B(x), \]

we obtain

\[ \boxed{\exists x(H(x)\land\neg B(x))}. \]

Thus, “Not all heroes are brave” means that there exists at least one hero who is not brave.

Worked Example: Employees Working Overtime

Let

\[ E(x):x\text{ is an employee}, \]

\[ O(x):x\text{ works overtime}, \]

\[ A(x):x\text{ is awarded}. \]

The statement “Employee who work overtime are awarded” can be represented as

\[ \boxed{\forall x((E(x)\land O(x))\rightarrow A(x))}. \]

Worked Example: Nested Quantifier Translation

Let

\[ S(x):x\text{ is a student}, \]

\[ B(y):y\text{ is a book}, \]

\[ R(x,y):x\text{ has read }y. \]

“Every student has read some book” is

\[ \forall x\left(S(x)\rightarrow \exists y(B(y)\land R(x,y))\right). \]

But “There is a book that every student has read” is

\[ \exists y\left(B(y)\land \forall x(S(x)\rightarrow R(x,y))\right). \]

The two sentences are not equivalent. In the first, different students may have read different books. In the second, one particular book must have been read by every student.

Rules of Inferences for Propositional Logic

Definition

A rule of inference is a valid argument pattern that allows a conclusion to be derived from one or more premises.

Modus Ponens

\[ \frac{p,\quad p\rightarrow q}{\therefore q} \]

If \(p\) is true and \(p\rightarrow q\) is true, then \(q\) must be true.

Modus Tollens

\[ \frac{p\rightarrow q,\quad\neg q}{\therefore\neg p} \]

If \(p\) would imply \(q\), but \(q\) is false, then \(p\) must also be false.

Hypothetical Syllogism

\[ \frac{p\rightarrow q,\quad q\rightarrow r}{\therefore p\rightarrow r} \]

Disjunctive Syllogism

\[ \frac{p\lor q,\quad\neg p}{\therefore q} \]

Addition

\[ \frac{p}{\therefore p\lor q} \]

Simplification

\[ \frac{p\land q}{\therefore p} \]

Conjunction

\[ \frac{p,\quad q}{\therefore p\land q} \]

Resolution

\[ \frac{p\lor q,\quad\neg p\lor r}{\therefore q\lor r} \]

Worked Example: Hypothetical Syllogism and Modus Tollens

Premises:

1. If the program compiles, then it executes. \(p\rightarrow q\)
2. If the program executes, then output is produced. \(q\rightarrow r\)
3. No output is produced. \(\neg r\)

Conclusion: The program did not compile, \(\neg p\).

Derivation:

1. \(p\rightarrow q\) — premise.
2. \(q\rightarrow r\) — premise.
3. \(\neg r\) — premise.
4. \(p\rightarrow r\) — from 1 and 2 by hypothetical syllogism.
5. \(\neg p\) — from 4 and 3 by modus tollens.

Therefore the argument is valid.

Worked Example: Modus Ponens

Suppose

\[ p:\text{ The user enters the correct password}, \]

\[ q:\text{ Access is granted}. \]

Given

\[ p\rightarrow q \]

and

\[ p, \]

modus ponens gives

\[ q. \]

Valid Arguments in Propositional Logic

Definition

An argument consists of premises followed by a conclusion. An argument is valid if there is no possible situation in which all premises are true and the conclusion is false.

Argument Form

If the premises are

\[ p_1,p_2,\ldots,p_n \]

and the conclusion is \(q\), the argument is valid exactly when

\[ (p_1\land p_2\land\cdots\land p_n)\rightarrow q \]

is a tautology.

Method 1: Using Rules of Inference

One can derive the conclusion from the premises using established rules such as modus ponens, modus tollens, and hypothetical syllogism.

Method 2: Using a Truth Table

Construct a truth table and examine every row where all premises are true. If the conclusion is also true in every such row, the argument is valid.

Worked Example

Premises:

\[ p\rightarrow q, \]

\[ q\rightarrow r, \]

\[ p. \]

Conclusion:

\[ r. \]

Derivation:

1. \(p\rightarrow q\) — premise.
2. \(q\rightarrow r\) — premise.
3. \(p\) — premise.
4. \(q\) — from 1 and 3 by modus ponens.
5. \(r\) — from 2 and 4 by modus ponens.

Alternatively, 1 and 2 give \(p\rightarrow r\) by hypothetical syllogism, and then 3 gives \(r\) by modus ponens.

Validity Versus Truth

Validity concerns the structure of reasoning, not whether every premise is factually true.

For example:

• All birds are mammals.
• All mammals are robots.
• Therefore, all birds are robots.

The premises are factually false, but the argument pattern is logically valid because the conclusion follows from those premises.

Fallacies

Definition

A fallacy is an invalid pattern of reasoning that may resemble a valid inference rule.

Fallacy of Affirming the Consequent

The invalid pattern is

\[ \frac{p\rightarrow q,\quad q}{\therefore p}. \]

This is sometimes informally described as affirming the conclusion.

Example

Suppose:

• If it is raining, then the road is wet.
• The road is wet.
• Therefore, it is raining.

Symbolically:

\[ p\rightarrow q,\quad q,\quad\therefore p. \]

The reasoning is invalid because the road might be wet for another reason, such as cleaning or a burst pipe.

Fallacy of Denying the Antecedent

The invalid form is

\[ \frac{p\rightarrow q,\quad\neg p}{\therefore\neg q}. \]

This is sometimes described as denying the hypothesis.

Example

Suppose:

• If I study for ten hours, then I pass the test.
• I did not study for ten hours.
• Therefore, I did not pass the test.

This conclusion does not necessarily follow. The person might pass after studying for fewer hours.

Comparison with Valid Rules

How to Show a Fallacy Is Invalid

To disprove an argument form, it is enough to find one assignment in which all premises are true but the conclusion is false.

For affirming the consequent, take

\[ p=F,\qquad q=T. \]

Then \(p\rightarrow q=T\) and \(q=T\), but the claimed conclusion \(p\) is false. Therefore the form is invalid.

Rules of Inferences for Quantified Statements

Definition

Rules of inference for quantified statements allow conclusions to be derived from statements containing universal and existential quantifiers.

Universal Instantiation

From

\[ \forall x\,P(x), \]

we may infer

\[ P(c) \]

for any particular element \(c\) in the domain.

The rule is written

\[ \frac{\forall x\,P(x)}{\therefore P(c)}. \]

Universal Generalization

If \(P(c)\) has been proved for an arbitrary element \(c\) of the domain without using any special property of \(c\), then we may conclude

\[ \forall x\,P(x). \]

Existential Instantiation

From

\[ \exists x\,P(x), \]

we may introduce a new symbol \(c\) representing some element for which \(P(c)\) is true.

The element must be treated as an unspecified witness rather than as an arbitrarily chosen object.

Existential Generalization

From a specific true statement

\[ P(c), \]

we may conclude

\[ \exists x\,P(x). \]

Comparison

Worked Example: Universal Instantiation with Modus Ponens

Premises:

1. Every computer science student studies logic.
2. Maya is a computer science student.

Conclusion: Maya studies logic.

Define

\[ C(x):x\text{ is a computer science student}, \]

\[ L(x):x\text{ studies logic}. \]

The first premise is

\[ \forall x(C(x)\rightarrow L(x)). \]

The second premise is

\[ C(\text{Maya}). \]

By universal instantiation,

\[ C(\text{Maya})\rightarrow L(\text{Maya}). \]

Using modus ponens with \(C(\text{Maya})\), we obtain

\[ L(\text{Maya}). \]

Therefore, Maya studies logic.

Valid Arguments in Quantified Statements

Definition

A quantified argument is valid when its conclusion follows logically from its quantified premises according to valid inference rules.

Worked Example

Premises:

1. All programmers understand algorithms.
2. Anyone who understands algorithms can solve certain computational problems.
3. Rita is a programmer.

Conclusion: Rita can solve certain computational problems.

Define

\[ P(x):x\text{ is a programmer}, \]

\[ A(x):x\text{ understands algorithms}, \]

\[ C(x):x\text{ can solve the relevant computational problems}. \]

The premises become

\[ \forall x(P(x)\rightarrow A(x)), \]

\[ \forall x(A(x)\rightarrow C(x)), \]

\[ P(\text{Rita}). \]

By universal instantiation,

\[ P(\text{Rita})\rightarrow A(\text{Rita}), \]

and

\[ A(\text{Rita})\rightarrow C(\text{Rita}). \]

Using \(P(\text{Rita})\) and modus ponens,

\[ A(\text{Rita}). \]

Using modus ponens again,

\[ C(\text{Rita}). \]

Therefore the argument is valid.

Example Involving Existence

Suppose

\[ \exists x(P(x)\land Q(x)). \]

Then some object \(c\) satisfies

\[ P(c)\land Q(c). \]

By simplification,

\[ P(c). \]

By existential generalization,

\[ \exists xP(x). \]

Therefore,

\[ \exists x(P(x)\land Q(x)) \rightarrow \exists xP(x) \]

is a valid logical implication.

Methods of Proving Theorems

Introduction

A mathematical proof is a logically valid argument that establishes a mathematical statement from definitions, axioms, previously proved results, and accepted inference rules. Different theorem forms suggest different proof techniques.

Direct Proof

Definition

A direct proof of an implication

\[ p\rightarrow q \]

begins by assuming the hypothesis \(p\) and then uses definitions and valid deductions to establish \(q\).

Worked Example

Theorem: The sum of two even integers is even.

Proof: Let \(m\) and \(n\) be even integers. By definition of even integer, there exist integers \(a\) and \(b\) such that

\[ m=2a,\qquad n=2b. \]

Then

\[ m+n=2a+2b=2(a+b). \]

Because \(a+b\in\mathbb{Z}\), \(m+n\) is twice an integer. Hence \(m+n\) is even.

Indirect Proof Using Contrapositive

Definition

To prove

\[ p\rightarrow q, \]

one may instead prove its logically equivalent contrapositive

\[ \neg q\rightarrow\neg p. \]

Worked Example

Theorem: If \(n^2\) is even, then \(n\) is even.

Contrapositive: If \(n\) is odd, then \(n^2\) is odd.

Assume \(n\) is odd. Then for some \(k\in\mathbb{Z}\),

\[ n=2k+1. \]

Therefore,

\[ n^2=(2k+1)^2 =4k^2+4k+1 =2(2k^2+2k)+1. \]

Since \(2k^2+2k\) is an integer, \(n^2\) is odd.

Thus the contrapositive is true, so the original statement is true.

Proof by Contradiction

Definition

To prove a statement \(P\) by contradiction, assume its negation \(\neg P\) and derive an impossibility. Since the assumption leads to contradiction, \(P\) must be true.

Worked Example: Proving \(\sqrt{2}\) Is Irrational

Theorem:

\[ \sqrt{2} \]

is irrational.

Proof by contradiction:

Assume the opposite, that \(\sqrt{2}\) is rational. Then it can be expressed as

\[ \sqrt{2}=\frac{a}{b}, \]

where \(a,b\in\mathbb{Z}\), \(b\neq0\), and the fraction is in lowest terms, so \(\gcd(a,b)=1\).

Squaring gives

\[ 2=\frac{a^2}{b^2}. \]

Hence,

\[ a^2=2b^2. \]

Therefore \(a^2\) is even. Since an integer whose square is even must itself be even, \(a\) is even.

Thus for some integer \(k\),

\[ a=2k. \]

Substituting,

\[ (2k)^2=2b^2, \]

so

\[ 4k^2=2b^2. \]

Dividing by \(2\),

\[ b^2=2k^2. \]

Therefore \(b^2\) is even, and hence \(b\) is even.

Thus both \(a\) and \(b\) are even. They therefore have a common factor \(2\), contradicting the original assumption that

\[ \gcd(a,b)=1. \]

Therefore the assumption that \(\sqrt2\) is rational is false. Hence

\[ \boxed{\sqrt2\text{ is irrational}.} \]

Worked Example: Sum of Two Odd Integers

The supplied exam wording states “the sum of odd integers is even.” Taken literally, this is not true for an arbitrary number of odd integers: for example \(1+3+5=9\) is odd. The standard intended theorem is that the sum of two odd integers is even.

Let

\[ m=2a+1,\qquad n=2b+1 \]

be odd integers. Suppose, for contradiction, that \(m+n\) is odd.

But

\[ m+n=(2a+1)+(2b+1) =2(a+b+1), \]

which is even.

This contradicts the assumption that \(m+n\) is odd. Therefore the sum of two odd integers is even.

Vacuous Proof

Definition

A conditional statement

\[ p\rightarrow q \]

is automatically true whenever \(p\) is false. A proof that establishes the impossibility of the hypothesis is called a vacuous proof.

Worked Example

Statement: If a real number \(x\) satisfies both

\[ x>5 \]

and

\[ x<2, \]

then \(x^2=10\).

No real number satisfies both \(x>5\) and \(x<2\). Therefore the hypothesis is always false.

Hence the implication is vacuously true, regardless of the conclusion.

Trivial Proof

Definition

A conditional statement \(p\rightarrow q\) is automatically true whenever \(q\) is always true. Establishing the truth of the conclusion independently of the hypothesis is called a trivial proof.

Worked Example

Statement: If an integer \(n\) is divisible by \(10\), then

\[ n^2\geq0. \]

For every integer \(n\), regardless of divisibility,

\[ n^2\geq0. \]

Thus the conclusion is always true, so the implication is trivially true.

Proof of Equivalence

Definition

To prove

\[ p\leftrightarrow q, \]

we must prove both directions:

\[ p\rightarrow q \]

and

\[ q\rightarrow p. \]

Worked Example

Theorem: An integer \(n\) is even if and only if \(n^2\) is even.

Direction 1: If \(n\) is even, then \(n^2\) is even

If \(n=2k\), then

\[ n^2=(2k)^2=4k^2=2(2k^2), \]

so \(n^2\) is even.

Direction 2: If \(n^2\) is even, then \(n\) is even

This was proved above using the contrapositive. If \(n\) were odd, then \(n^2\) would be odd. Therefore \(n^2\) even implies \(n\) even.

Hence,

\[ n\text{ is even} \leftrightarrow n^2\text{ is even}. \]

Exhaustive Proof

Definition

An exhaustive proof checks every possible case individually. It is practical when the domain contains only a small finite number of possibilities.

Worked Example

Theorem: For every

\[ n\in\{1,2,3,4\}, \]

the number \(n^2+n\) is even.

Check all possible values:

• \(n=1:\quad1^2+1=2\), even.
• \(n=2:\quad2^2+2=6\), even.
• \(n=3:\quad3^2+3=12\), even.
• \(n=4:\quad4^2+4=20\), even.

Since every possible value in the stated domain has been checked, the theorem is proved.

Proof by Cases

Definition

In a proof by cases, the domain is divided into cases that together cover every possibility. The statement is then proved separately in each case.

Worked Example

Theorem: For every integer \(n\),

\[ n(n+1) \]

is even.

Every integer is either even or odd.

Case 1: \(n\) is even

If

\[ n=2k, \]

then

\[ n(n+1)=2k(n+1), \]

which is even.

Case 2: \(n\) is odd

If

\[ n=2k+1, \]

then

\[ n+1=2k+2=2(k+1), \]

so

\[ n(n+1)=n\cdot2(k+1), \]

which is even.

Since the expression is even in both possible cases, \(n(n+1)\) is always even.

Existence Proof

Definition

An existence proof establishes that at least one object satisfying specified conditions exists.

A constructive existence proof explicitly provides such an object.

Worked Example

Claim: There exist positive integers \(x\) and \(y\) such that

\[ x^2+y^2=25. \]

Choose

\[ x=3,\qquad y=4. \]

Then

\[ 3^2+4^2=9+16=25. \]

Thus such positive integers exist.

Uniqueness Proof

Definition

A uniqueness proof shows not only that an object exists but also that no second distinct object can satisfy the same conditions.

A common structure is:

1. prove existence;
2. assume two objects satisfy the condition;
3. show that the two objects must be equal.

Worked Example

Theorem: For real numbers \(a\neq0\) and \(b\), there exists a unique real number \(x\) satisfying

\[ ax+b=0. \]

Existence

Choose

\[ x=-\frac{b}{a}. \]

Then

\[ a\left(-\frac{b}{a}\right)+b=-b+b=0. \]

So a solution exists.

Uniqueness

Suppose \(x_1\) and \(x_2\) are both solutions. Then

\[ ax_1+b=0 \]

and

\[ ax_2+b=0. \]

Subtracting gives

\[ a(x_1-x_2)=0. \]

Since \(a\neq0\),

\[ x_1-x_2=0. \]

Therefore,

\[ x_1=x_2. \]

Hence the solution is unique.

Counter Example

Definition

A counterexample is a specific example showing that a universal statement is false.

To disprove

\[ \forall x\,P(x), \]

it is sufficient to find one \(x\) for which \(P(x)\) is false.

Worked Example

Claim: For every integer \(n\),

\[ n^2>n. \]

Take

\[ n=1. \]

Then

\[ 1^2=1, \]

which is not greater than \(1\).

Therefore the universal claim is false.

Mistakes in Proof

Definition

A proof is valid only when every step follows logically from definitions, assumptions, axioms, or previously established results. A mistake in proof occurs when an invalid algebraic operation, unjustified inference, hidden assumption, circular argument, or incomplete case analysis is used.

Common Proof Mistakes

• Dividing by zero: cancelling a factor that may equal zero.
• Assuming what must be proved: using the conclusion as a premise, producing circular reasoning.
• Proving only the converse: proving \(q\rightarrow p\) when the theorem asks for \(p\rightarrow q\).
• Using examples as a universal proof: checking several examples does not prove a statement for every possible value unless the domain is completely exhausted.
• Ignoring a case: a proof by cases is incomplete if the cases do not cover all possibilities.
• Invalid quantifier reasoning: treating “there exists” as “for every” or generalizing from a special example.

Worked Example: Faulty Proof that \(1=2\)

Consider the following argument.

Assume

\[ a=b. \]

Multiply both sides by \(a\):

\[ a^2=ab. \]

Subtract \(b^2\):

\[ a^2-b^2=ab-b^2. \]

Factor:

\[ (a-b)(a+b)=b(a-b). \]

Cancel \(a-b\):

\[ a+b=b. \]

Since \(a=b\),

\[ 2b=b. \]

Dividing by \(b\) seems to give

\[ 2=1. \]

Finding the Error

Because the original assumption was \(a=b\),

\[ a-b=0. \]

The step that cancels \(a-b\) therefore divides both sides by zero. Division by zero is undefined.

Hence the argument is invalid.

Example of Circular Reasoning

Suppose a person tries to prove that the sum of two even integers is even by writing:

“The sum is even because sums of even integers are even.”

This merely restates the theorem and gives no justification. The correct proof must use the definition \(m=2a\), \(n=2b\) and derive \(m+n=2(a+b)\).

Example of Proving the Wrong Direction

Suppose the theorem is

\[ p\rightarrow q. \]

Proving only

\[ q\rightarrow p \]

does not prove the original theorem unless the reverse implication is independently known to be equivalent in that context.

Exam Questions

1. Why do we need quantifier?
   (Mar 2024, Q1 / Nov 2024, Q1)
2. Define logic.
   (Mar 2024, Q7 / Nov 2024, Q7)
3. Using proof by contradiction, show that the sum of odd integers is even.
   Editorial note: wording preserved as written; as explained above, the unrestricted statement is ambiguous because the parity depends on how many odd integers are being summed.
   (Mar 2024, Q18 / Nov 2024, Q18)
4. Represent the following sentences using predicate.
   a. Not all heroes are brave.
   b. Employee who work overtime are awarded.
   (Mar 2024, Q20 / Nov 2024, Q20)
5. State inference rule for universal instantiation.
   (2023, Q1)
6. Give an example of fallacy.
   (2023, Q4)
7. List the negation of quantifier with example. Prove that \(\sqrt{2}\) is irrational.
   Editorial note: the source does not name the proof method; the standard proof is by contradiction. The negation-of-quantifiers part belongs to Predicate and Quantifiers.
   (2023, Q21)

Whole Chapter Summary

Propositional logic: Logic studies valid reasoning. Propositions have definite truth values, and compound propositions are formed from simple propositions using logical connectives.

Logical operators: NOT negates a statement, AND requires both components, OR requires at least one, implication fails only in the true-to-false case, and biconditional requires equal truth values. An implication is equivalent to its contrapositive.

Logical equivalence: Equivalent propositions have identical truth values for all valuations. Important tools include De Morgan's laws, implication laws, distributive laws, absorption, and double negation.

English translation: Natural-language statements can be represented using propositional variables and logical operators. Special attention is required for expressions such as “if,” “only if,” and “unless.”

Predicates and quantifiers: A predicate contains variables, while quantifiers bind those variables. Universal quantification expresses “for all,” existential quantification expresses “there exists,” and negating one quantifier changes it into the other.

Nested quantifiers: Quantifier order matters when \(\forall\) and \(\exists\) are mixed. \(\forall x\exists y\) usually allows the witness \(y\) to depend on \(x\), whereas \(\exists y\forall x\) requires one witness for all \(x\).

Quantified translations: “Every” typically uses a universally quantified implication, while “some” typically uses an existentially quantified conjunction. “Not all” expresses the existence of an exception.

Rules of inference: Modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, conjunction, simplification, addition, and resolution are standard valid propositional argument patterns.

Valid arguments: An argument is valid when no possible interpretation makes all premises true and its conclusion false. Validity can be checked by formal derivation or truth tables.

Fallacies: Affirming the consequent and denying the antecedent resemble valid rules but do not preserve truth.

Quantified inference: Universal instantiation, universal generalization, existential instantiation, and existential generalization allow reasoning involving quantified predicates.

Proof methods: Mathematical statements may be proved directly, through a contrapositive, by contradiction, vacuously, trivially, through equivalence, by exhaustive checking, by cases, through existence or uniqueness arguments, or disproved with counterexamples.

Mistakes in proof: Invalid cancellation, circular reasoning, proving the converse instead of the original implication, incomplete cases, and invalid quantifier generalization can invalidate an otherwise convincing-looking argument.

Key Formulas and Points Sheet

Last-Minute Revision Sheet