Class 9 Mathematics Notes – Chapter 8: Logic

Class: 9th
Subject: Mathematics
Chapter: 08 – Logic
Board: Punjab Board
Exercise: 8

Q1. Multiple Choice Questions (MCQs)

(i) Which of the following expressions is often related to inductive reasoning?

Answer: (a) Based on repeated experiments

Reasoning: Inductive reasoning derives general rules or conclusions from repeated observations or experimental trials.

(ii) Which of the following sentences describes deductive reasoning?

Answer: (d) Draw conclusion from well-known facts

Reasoning: Deductive reasoning uses logical steps to draw necessary conclusions from accepted or known true statements (premises).

(iii) Which one of the following statements is true?

Answer: (c) \(\frac{22}{7}\)

Reasoning: \(\frac{22}{7}\) is a rational number because it is the quotient of two integers. Therefore, \(\frac{22}{7}\notin\mathbb{Q}’\), where \(\mathbb{Q}’\) represents irrational numbers.

(iv) Which of the following statements best represents the negation of “The stove is burning”?

Answer: (a) The stove is not burning.

Reasoning: The negation of a proposition \(p\) is written as \(\sim p\), meaning “not \(p\).”

(v) The conjunction of two statements \(p\) and \(q\) is true when:

Answer: (b) Both \(p\) and \(q\) are true.

Reasoning: The conjunction \(p\land q\) is true only when both statements are true.

(vi) A conditional statement is regarded as false only when:

Answer: (a) The antecedent is true and the consequent is false.

Reasoning: For \(p\rightarrow q\), the combination \(T\rightarrow F\) is the only false case in its truth table.

(vii) The contrapositive of \(q\rightarrow p\) is:

Answer: (b) \(\sim p\rightarrow\sim q\)

Reasoning: To form the contrapositive, reverse the direction of the conditional and negate both statements.

(viii) The statement “Every even integer greater than 2 is a sum of two prime numbers” is:

Answer: (b) Conjecture

Reasoning: This is known as Goldbach’s conjecture. It remains unproven mathematically.

(ix) The statement “A straight line can be drawn between any two points” is:

Answer: (c) Axiom

Reasoning: This is accepted as a basic geometric postulate without requiring proof.

(x) The statement “The sum of the interior angles of a triangle is \(180^\circ\)” is:

Answer: (b) Theorem

Reasoning: It is a proven mathematical result established through geometric reasoning.

Q2. Write the Converse, Inverse, and Contrapositive of the Following Conditionals

For a conditional statement \(A\rightarrow B\):

Converse: \(B\rightarrow A\)
Inverse: \(\sim A\rightarrow\sim B\)
Contrapositive: \(\sim B\rightarrow\sim A\)

(i) \(\sim p\rightarrow q\)

Converse: \(q\rightarrow\sim p\)
Inverse: \(p\rightarrow\sim q\)
Contrapositive: \(\sim q\rightarrow p\)

(ii) \(q\rightarrow p\)

Converse: \(p\rightarrow q\)
Inverse: \(\sim q\rightarrow\sim p\)
Contrapositive: \(\sim p\rightarrow\sim q\)

(iii) \(\sim p\rightarrow\sim q\)

Converse: \(\sim q\rightarrow\sim p\)
Inverse: \(p\rightarrow q\)
Contrapositive: \(q\rightarrow p\)

(iv) \(\sim q\rightarrow p\)

Converse: \(p\rightarrow\sim q\)
Inverse: \(q\rightarrow\sim p\)
Contrapositive: \(\sim p\rightarrow q\)

Q3. Write the Truth Tables of the Following

(i) \(\sim(p\lor q)\lor(\sim q)\)

\(p\)	\(q\)	\(p\lor q\)	\(\sim(p\lor q)\)	\(\sim q\)	\(\sim(p\lor q)\lor(\sim q)\)
T	T	T	F	F	F
T	F	T	F	T	T
F	T	T	F	F	F
F	F	F	T	T	T

(ii) \(\sim(\sim q\lor\sim p)\)

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(\sim q\lor\sim p\)	\(\sim(\sim q\lor\sim p)\)
T	T	F	F	F	T
T	F	F	T	T	F
F	T	T	F	T	F
F	F	T	T	T	F

Note: This expression is logically equivalent to \(p\land q\).

(iii) \((p\lor q)\leftrightarrow(p\land q)\)

\(p\)	\(q\)	\(p\lor q\)	\(p\land q\)	\((p\lor q)\leftrightarrow(p\land q)\)
T	T	T	T	T
T	F	T	F	F
F	T	T	F	F
F	F	F	F	T

Q4. Differentiate Between a Mathematical Statement and Its Proof. Give Two Examples.

Mathematical Statement: A declarative sentence or expression that is either true or false, but not both.

Examples:

The sum of the interior angles of a triangle is \(180^\circ\).
If \(x\) is an odd integer, then \(x^2\) is an odd integer.

Proof: A logical sequence of valid arguments, definitions, axioms, and previously established theorems that demonstrates the truth of a mathematical statement.

Example 1: Prove that the square of an odd integer is odd.

Let \(x=2k+1\), where \(k\in\mathbb Z\).

\[ \begin{aligned} x^2&=(2k+1)^2\\ &=4k^2+4k+1\\ &=2(2k^2+2k)+1 \end{aligned} \]

Let \(m=2k^2+2k\), where \(m\in\mathbb Z\).

Therefore,

\[ x^2=2m+1 \]

Hence, \(x^2\) is odd.

Example 2: Prove that the sum of the interior angles of a triangle is \(180^\circ\).

Draw a line parallel to one side of the triangle through the opposite vertex. Using the properties of alternate interior angles and angles on a straight line, the sum of the three interior angles is \(180^\circ\).

Q5. What Is the Difference Between an Axiom and a Theorem? Give Examples.

Axiom: A fundamental statement accepted as true without proof and used as a starting point for further reasoning.

Example: Through any two distinct points, exactly one straight line can be drawn.

Theorem: A mathematical statement whose truth is established through logical proof based on axioms, definitions, and previously proved theorems.

Example: Pythagoras’ theorem states that in a right-angled triangle,

\[ a^2+b^2=c^2 \]

where \(c\) is the hypotenuse.

Q6. What Is the Importance of Logical Reasoning in Mathematical Proofs? Give an Example.

Importance: Logical reasoning ensures that mathematical conclusions follow valid steps and accepted rules. It helps prevent false assumptions and distinguishes a proven result from a conclusion based only on observations.

Example: Consider the expression

\[ n^2+n+41 \]

It produces prime numbers for \(n=0,1,2,\ldots,39\). However, this does not mean the expression always produces a prime number.

For \(n=41\),

\[ \begin{aligned} n^2+n+41&=41^2+41+41\\ &=41(41+1+1)\\ &=41\times43 \end{aligned} \]

Since the result is a product of two integers greater than 1, it is composite.

Therefore, repeated observations alone are not sufficient to establish a mathematical statement as universally true.

Q7. Indicate Whether Each Statement Is an Axiom, Conjecture, or Theorem. Explain Your Reasoning.

(i) There is exactly one straight line through any two distinct points.

Classification: Axiom

Reasoning: It is a basic geometric principle accepted without proof.

(ii) Every even number greater than 2 can be written as the sum of two prime numbers.

Classification: Conjecture

Reasoning: This is known as Goldbach’s conjecture. Despite extensive supporting evidence, it has not been proved in general.

(iii) The sum of the interior angles of a triangle is \(180^\circ\).

Classification: Theorem

Reasoning: It can be proved using geometric properties involving parallel lines and angles.

Q8. Formulate Simple Deductive Proofs for the Following Algebraic Expressions. Prove That LHS = RHS.

(i) Prove that \((x-4)^2+9=x^2-8x+25\)

Proof:

\[ \begin{aligned} \text{LHS}&=(x-4)^2+9\\ &=x^2-2(x)(4)+4^2+9\\ &=x^2-8x+16+9\\ &=x^2-8x+25\\ &=\text{RHS} \end{aligned} \]

Hence proved.

(ii) Prove that \((x+1)^2-(x-1)^2=4x\)

Proof:

\[ \begin{aligned} \text{LHS}&=(x+1)^2-(x-1)^2\\ &=(x^2+2x+1)-(x^2-2x+1)\\ &=x^2+2x+1-x^2+2x-1\\ &=4x\\ &=\text{RHS} \end{aligned} \]

Hence proved.

(iii) Prove that \((x+5)^2-(x-5)^2=20x\)

Proof:

\[ \begin{aligned} \text{LHS}&=(x+5)^2-(x-5)^2\\ &=(x^2+10x+25)-(x^2-10x+25)\\ &=x^2+10x+25-x^2+10x-25\\ &=20x\\ &=\text{RHS} \end{aligned} \]

Hence proved.

Q9. Prove the Following by Justifying Each Step.

(i) Prove that \(\frac{4+16x}{4}=1+4x\)

Proof:

\[ \begin{aligned} \text{LHS}&=\frac{4+16x}{4}\\ &=\frac{4}{4}+\frac{16x}{4}\\ &=1+4x\\ &=\text{RHS} \end{aligned} \]

Hence proved.

(ii) Verify the Following Identity

\[ \frac{6x^2+18x}{3x^2-9}=\frac{2x}{x-3} \]

Note: The supplied question appears to contain an error. As written, the identity is not generally true.

The left-hand side factors as

\[ \begin{aligned} \frac{6x^2+18x}{3x^2-9} &=\frac{6x(x+3)}{3(x^2-3)}\\ &=\frac{2x(x+3)}{x^2-3} \end{aligned} \]

This does not simplify to \(\frac{2x}{x-3}\). Please check the original textbook question before publishing this solution.

(iii) Prove that

\[ \frac{x^2+7x+10}{x^2-3x-10} =\frac{x+5}{x-5} \]

Proof:

Factorize the numerator and denominator:

\[ x^2+7x+10=(x+5)(x+2) \]

\[ x^2-3x-10=(x+2)(x-5) \]

Therefore,

\[ \begin{aligned} \text{LHS} &=\frac{(x+5)(x+2)}{(x+2)(x-5)}\\ &=\frac{x+5}{x-5}\\ &=\text{RHS} \end{aligned} \]

Hence proved, for \(x\ne -2,5\).

Q10. Suppose \(x\) Is an Integer. Then \(x\) Is Odd If and Only If \(9x+4\) Is Odd.

Proof:

Part 1: If \(x\) is odd, then \(9x+4\) is odd.

Let

\[ x=2k+1,\qquad k\in\mathbb Z \]

Substituting into \(9x+4\),

\[ \begin{aligned} 9x+4&=9(2k+1)+4\\ &=18k+9+4\\ &=18k+13\\ &=2(9k+6)+1 \end{aligned} \]

Since \(9k+6\) is an integer, \(9x+4\) is odd.

Part 2: If \(9x+4\) is odd, then \(x\) is odd.

We prove this by contrapositive.

Assume that \(x\) is even. Then

\[ x=2k,\qquad k\in\mathbb Z \]

Substituting,

\[ \begin{aligned} 9x+4&=9(2k)+4\\ &=18k+4\\ &=2(9k+2) \end{aligned} \]

Since \(9k+2\) is an integer, \(9x+4\) is even.

Thus, if \(x\) is even, \(9x+4\) is even. By contrapositive, if \(9x+4\) is odd, \(x\) must be odd.

Conclusion: \(x\) is odd if and only if \(9x+4\) is odd.

Q11. Suppose \(x\) Is an Integer. If \(x\) Is Odd, Then \(7x+5\) Is Even.

Proof:

Let \(x\) be an odd integer. By definition,

\[ x=2k+1,\qquad k\in\mathbb Z \]

Substituting into \(7x+5\),

\[ \begin{aligned} 7x+5&=7(2k+1)+5\\ &=14k+7+5\\ &=14k+12\\ &=2(7k+6) \end{aligned} \]

Since \(7k+6\) is an integer, let \(m=7k+6\).

Then,

\[ 7x+5=2m \]

Therefore, \(7x+5\) is even.

Hence proved.

Q12. Prove the Following Statements.

(a) If \(x\) Is an Odd Integer, Then \(x^2-4x+6\) Is Odd.

Proof:

Let \(x\) be an odd integer. Then

\[ x=2k+1,\qquad k\in\mathbb Z \]

Substituting into the expression,

\[ \begin{aligned} x^2-4x+6 &=(2k+1)^2-4(2k+1)+6\\ &=4k^2+4k+1-8k-4+6\\ &=4k^2-4k+3\\ &=2(2k^2-2k+1)+1 \end{aligned} \]

Since \(2k^2-2k+1\) is an integer, the expression is of the form \(2m+1\).

Therefore, \(x^2-4x+6\) is odd.

Hence proved.

Notes by Sir Dilawar

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