Review Exercise 8
Review Exercise 8 Solutions
Complete solved answers for mathematical logic, conditionals, truth tables, deductive proofs, and algebraic properties.
Question 1
Interactive logic Quiz
Choose the correct option. Test your understanding of inductive reasoning, propositions, and connectives using the tabs below.
Question 2
Converse, Inverse, Contrapositive
Write the converse, inverse and contrapositive of the following conditionals:
(i) ~p ⇒ q (ii) q ⇒ p (iii) ~p ⇒ ~q (iv) ~q ⇒ ~p
(i) ~p ⇒ q (ii) q ⇒ p (iii) ~p ⇒ ~q (iv) ~q ⇒ ~p
Solutions
(i) ~p ⇒ q:
• Converse (q ⇒ p): q ⇒ ~p
• Inverse (~p ⇒ ~q): p ⇒ ~q
• Contrapositive (~q ⇒ ~p): ~q ⇒ p
• Converse (q ⇒ p): q ⇒ ~p
• Inverse (~p ⇒ ~q): p ⇒ ~q
• Contrapositive (~q ⇒ ~p): ~q ⇒ p
(ii) q ⇒ p:
• Converse: p ⇒ q
• Inverse: ~q ⇒ ~p
• Contrapositive: ~p ⇒ ~q
• Converse: p ⇒ q
• Inverse: ~q ⇒ ~p
• Contrapositive: ~p ⇒ ~q
(iii) ~p ⇒ ~q:
• Converse: ~q ⇒ ~p
• Inverse: p ⇒ q
• Contrapositive: q ⇒ p
• Converse: ~q ⇒ ~p
• Inverse: p ⇒ q
• Contrapositive: q ⇒ p
(iv) ~q ⇒ ~p:
• Converse: ~p ⇒ ~q
• Inverse: q ⇒ p
• Contrapositive: p ⇒ q
• Converse: ~p ⇒ ~q
• Inverse: q ⇒ p
• Contrapositive: p ⇒ q
Question 3
Truth Tables
Write the truth tables of the following:
(i) ~(p ∨ q) ∨ (~q) (ii) ~(~q ∨ ~p) (iii) (q ∨ p) ⇒ (p ∧ q)
(i) ~(p ∨ q) ∨ (~q) (ii) ~(~q ∨ ~p) (iii) (q ∨ p) ⇒ (p ∧ q)
(i) ~(p ∨ q) ∨ (~q)
| p | q | p ∨ q | ~(p ∨ q) | ~q | ~(p ∨ q) ∨ (~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) ~(~q ∨ ~p)
| p | q | ~p | ~q | ~q ∨ ~p | ~(~q ∨ ~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: The final column matches the truth table for p ∧ q, verifying logical equivalence by De Morgan's Law.
(iii) (q ∨ p) ⇒ (p ∧ q)
| p | q | q ∨ p | p ∧ q | (q ∨ p) ⇒ (p ∧ q) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | T | F | F |
| F | T | T | F | F |
| F | F | F | F | T |
Question 4
Statement & Proof
Differentiate between a Mathematical Statement and its proof and provide two examples.
Comparative Analysis
Mathematical Statement: A sentence or logical expression which is either definitely true or definitely false, but not both. It represents a single unit of mathematical claim.
Example 1: The sum of the measures of the interior angles of a triangle is 180°.
Example 2: For a non-zero real number x and integers m, n: xm · xn = xm+n.
Example 1: The sum of the measures of the interior angles of a triangle is 180°.
Example 2: For a non-zero real number x and integers m, n: xm · xn = xm+n.
Mathematical Proof: A systematic, step-by-step logical argument that establishes the truth of a mathematical statement. It connects axioms, postulates, and previously proven theorems using rules of inference to demonstrate why the statement must always hold.
Triangle Angle Sum Proof: By drawing a line parallel to the base through the opposite vertex, alternate interior angles are shown to form a straight line of 180°, logically proving the statement.
Triangle Angle Sum Proof: By drawing a line parallel to the base through the opposite vertex, alternate interior angles are shown to form a straight line of 180°, logically proving the statement.
Question 5
Axiom vs Theorem
What is the difference between an axiom and a theorem? Provide examples of each.
Comparative Analysis
Axiom: A statement that is assumed to be true without proof. It serves as a starting point or foundation for building mathematical theories.
Example: "Through any two points in a plane, exactly one straight line can be drawn." (Euclid's First Postulate).
Example: "Through any two points in a plane, exactly one straight line can be drawn." (Euclid's First Postulate).
Theorem: A mathematical statement that has been proven true using deductive reasoning, starting from axioms and other established theorems.
Example: "The sum of the interior angles of a quadrilateral is 360°." (This requires proof by splitting the quadrilateral into two triangles).
Example: "The sum of the interior angles of a quadrilateral is 360°." (This requires proof by splitting the quadrilateral into two triangles).
Question 6
Logical Reasoning
What is the importance of logical reasoning in mathematical proofs? Give an example to illustrate your point.
Explanation & Example
Importance: Logical reasoning provides the objective framework that guarantees a mathematical claim is universally true. Unlike natural sciences which rely on induction (empirical observations and experiments), mathematics demands deduction to ensure that conclusions are absolute and free from counterexamples.
Example: Proving "The sum of two odd numbers is always even."
• We cannot just test cases like 3 + 5 = 8 (induction).
• Instead, let any two odd integers be represented as 2m + 1 and 2n + 1 (where m, n ∈ Z).
• Sum:
• Since the sum is a multiple of 2, it is by definition even. This deductive proof guarantees the statement is true for all integers.
• We cannot just test cases like 3 + 5 = 8 (induction).
• Instead, let any two odd integers be represented as 2m + 1 and 2n + 1 (where m, n ∈ Z).
• Sum:
Sum
=
(2m + 1) + (2n + 1)
=
2m + 2n + 2
=
2(m + n + 1)
• Since the sum is a multiple of 2, it is by definition even. This deductive proof guarantees the statement is true for all integers.
Question 7
Classify Statements
Indicate each of the following whether it is an axiom, conjecture, or theorem, and explain your reasoning:
(i) "There is exactly one straight line through any two points."
(ii) "Every even number greater than 2 can be written as the sum of two prime numbers."
(iii) "The sum of the angles in a triangle is 180 degrees."
(i) "There is exactly one straight line through any two points."
(ii) "Every even number greater than 2 can be written as the sum of two prime numbers."
(iii) "The sum of the angles in a triangle is 180 degrees."
Classifications
(i) Axiom: This statement is self-evident and accepted as true without mathematical proof. It is one of Euclid's basic geometric postulates.
(ii) Conjecture: This is the famous **Goldbach's Conjecture**. Although it has been verified for trillions of even numbers, it has not been mathematically proven for all integers, meaning it remains a conjecture.
(iii) Theorem: This is a theorem because it can be (and has been) proven true using axioms of parallel lines and triangles.
Question 8
Deductive Proofs
For each of the following algebraic expressions, prove that the LHS is equal to the RHS:
(i) (x - 4)² + 9 = x² - 8x + 25
(ii) (x + 1)² - (x - 1)² = 4x
(iii) (x + 5)² - (x - 5)² = 20x
(i) (x - 4)² + 9 = x² - 8x + 25
(ii) (x + 1)² - (x - 1)² = 4x
(iii) (x + 5)² - (x - 5)² = 20x
(i) (x - 4)² + 9 = x² - 8x + 25
Proof
Expand LHS using the identity (a - b)² = a² - 2ab + b²:
LHS
=
(x - 4)2 + 9
=
(x2 - 8x + 16) + 9
=
x2 - 8x + 25 = RHS
(ii) (x + 1)² - (x - 1)² = 4x
Proof
Expand both terms:
LHS
=
(x2 + 2x + 1) - (x2 - 2x + 1)
=
x2 + 2x + 1 - x2 + 2x - 1
=
4x = RHS
(iii) (x + 5)² - (x - 5)² = 20x
Proof
Expand terms:
LHS
=
(x2 + 10x + 25) - (x2 - 10x + 25)
=
x2 + 10x + 25 - x2 + 10x - 25
=
20x = RHS
Question 9
Justifying Steps
Prove the following by justifying each step:
(i) (4 + 16x) / 4 = 1 + 4x (ii) (6x² + 18x) / (3x² - 9) = 2x / (x - 3) (iii) (x² + 7x + 10) / (x² - 3x - 10) = (x + 5) / (x - 5)
(i) (4 + 16x) / 4 = 1 + 4x (ii) (6x² + 18x) / (3x² - 9) = 2x / (x - 3) (iii) (x² + 7x + 10) / (x² - 3x - 10) = (x + 5) / (x - 5)
(i) (4 + 16x) / 4 = 1 + 4x
| Expression / Step | Justification |
|---|---|
| LHS = (4 + 16x) · (1/4) | Definition of Division |
| = 4 · (1 + 4x) · (1/4) | Distributive / Factoring Law |
| = (1 + 4x) · [4 · (1/4)] | Commutative & Associative properties |
| = (1 + 4x) · 1 | Multiplicative Inverse property |
| = 1 + 4x = RHS | Multiplicative Identity property |
(ii) (6x² + 18x) / (3x² - 9) = 2x / (x - 3)
| Expression / Step | Justification |
|---|---|
| LHS = [6x(x + 3)] / [3(x² - 9)] | Distributive property (Factoring numerator & denominator) |
| = [3 · 2x(x + 3)] / [3(x + 3)(x - 3)] | Difference of squares: x² - 9 = (x - 3)(x + 3) |
| = [3(x + 3) · 2x] / [3(x + 3) · (x - 3)] | Commutative property |
| = [3(x + 3) / 3(x + 3)] · [2x / (x - 3)] | Definition of Multiplication |
| = 1 · [2x / (x - 3)] | Multiplicative Inverse property |
| = 2x / (x - 3) = RHS | Multiplicative Identity property |
(iii) (x² + 7x + 10) / (x² - 3x - 10) = (x + 5) / (x - 5)
| Expression / Step | Justification |
|---|---|
| LHS = (x² + 2x + 5x + 10) / (x² + 2x - 5x - 10) | Trinomial Splitting of Middle Term |
| = [(x + 2)(x + 5)] / [(x + 2)(x - 5)] | Distributive / Factoring property |
| = [(x + 2) / (x + 2)] · [(x + 5) / (x - 5)] | Definition of multiplication |
| = 1 · [(x + 5) / (x - 5)] | Multiplicative Inverse property |
| = (x + 5) / (x - 5) = RHS | Multiplicative Identity property |
Questions 10 - 12
Deductive Proofs on Integers
Q10. Prove: If x is odd, then 9x + 4 is odd.
Proof
Since x is odd, we can write x = 2k + 1 (where k ∈ Z).
Substitute into the expression:
Substitute into the expression:
9x + 4
=
9(2k + 1) + 4
=
18k + 9 + 4
=
18k + 13 = 18k + 12 + 1
=
2(9k + 6) + 1
Since k is an integer, 9k + 6 is also an integer. Thus, 2(9k + 6) is even.
Therefore, 2(9k + 6) + 1 is odd. Hence, 9x + 4 is odd.
Therefore, 2(9k + 6) + 1 is odd. Hence, 9x + 4 is odd.
Q11. Prove: If x is odd, then 7x + 5 is even.
Proof
Substitute x = 2k + 1:
7x + 5
=
7(2k + 1) + 5
=
14k + 7 + 5
=
14k + 12
=
2(7k + 6)
Since the expression is a multiple of 2, 7x + 5 is even.
Q12. Show that: (a) If x is odd, then x² - 4x + 6 is odd. (b) If x is even, then x² + 2x + 4 is even.
Proof (a)
Substitute x = 2k + 1:
x2 - 4x + 6
=
(2k + 1)2 - 4(2k + 1) + 6
=
(4k2 + 4k + 1) - 8k - 4 + 6
=
4k2 - 4k + 3 = 4k2 - 4k + 2 + 1
=
2(2k2 - 2k + 1) + 1 = Even + 1 = Odd
Proof (b)
For even x, substitute x = 2k:
x2 + 2x + 4
=
(2k)2 + 2(2k) + 4
=
4k2 + 4k + 4
=
2(2k2 + 2k + 2) = Even
Question 13
De Morgan's Law
Prove that for any two non-empty sets A and B, (A ∧ B)' = A' ∨ B' (or (A ∩ B)' = A' ∪ B').
Proof by Logical Equivalence
The set theory statement corresponds directly to the logical law:
~(p ∧ q)
≡
~p ∨ ~q
We construct the truth table to verify this equivalence:
Since the last two columns match exactly, De Morgan's Law is logically proven.
| p | q | ~p | ~q | p ∧ q | ~(p ∧ q) | ~p ∨ ~q |
|---|---|---|---|---|---|---|
| T | T | F | F | T | F | F |
| T | F | F | T | F | T | T |
| F | T | T | F | F | T | T |
| F | F | T | T | F | T | T |
Question 14
Inequality Proof
If x and y are positive real numbers and x² < y² then x < y.
Proof
Given: x² < y² where x, y > 0.
x2 - y2
<
0
Factorize:
(x - y)(x + y)
<
0
Since both x and y are positive, their sum x + y must be strictly positive (x + y > 0).
For the product of two terms to be negative, and one is positive, the other term must be negative:
For the product of two terms to be negative, and one is positive, the other term must be negative:
x - y
<
0
x
<
y
Hence, it is proved that x < y.
Question 15
Triangle Angle Sum
Prove that the sum of the interior angles of a triangle is 180 degrees.
Proof Steps
Given: A triangle ΔABC with interior angles ∠1, ∠2, and ∠3 at vertices A, B, and C.
Construction: Draw a straight line L parallel to base side AC, passing through vertex B. Let the angles formed on the straight line be ∠4 and ∠5.
Deduction:
1. Since the line is straight, the angles on it sum to 180°:
2. By alternate interior angle theorems for parallel lines:
3. Substituting these values back:
1. Since the line is straight, the angles on it sum to 180°:
∠4 + ∠2 + ∠5
=
180°
2. By alternate interior angle theorems for parallel lines:
∠4
=
∠1 and ∠5 = ∠3
3. Substituting these values back:
∠1 + ∠2 + ∠3
=
180°
Hence, the sum of interior angles of a triangle is 180°.
Question 16
Algebraic Rules
If a, b and c are non-zero real numbers prove that:
(a) a/b = c/d ⇒ ad = bc (b) (a/b) · (c/d) = ac/bd (c) a/b + c/b = (a+c)/b
(a) a/b = c/d ⇒ ad = bc (b) (a/b) · (c/d) = ac/bd (c) a/b + c/b = (a+c)/b
Proofs
(a) a/b = c/d ⇒ ad = bc:
Multiply both sides by bd (Multiplicative property of equality):
(a/b) · bd
=
(c/d) · bd
Using Associative and Multiplicative Inverse properties:
a · (1/b · b) · d
=
c · (1/d · d) · b
ad
=
bc
(b) (a/b) · (c/d) = ac/bd:
Using reciprocal rules:
(a/b) · (c/d)
=
(a · b-1) · (c · d-1)
By Commutative and Associative laws of multiplication:
=
(a · c) · (b-1 · d-1)
=
ac · (bd)-1 = ac/bd
(c) a/b + c/b = (a+c)/b:
Rewrite division as multiplication by inverse:
a/b + c/b
=
a · (1/b) + c · (1/b)
By Right Distributive property:
=
(a + c) · (1/b) = (a + c)/b
Interactive Sandbox
Dynamic Truth Table Generator
Construct truth tables dynamically. Select or type logical expressions to analyze truth values.