Logic Explorer

A colorful journey through mathematical logic for class 10 students

Exercise 1

Let p: Jupiter is a planet and q: India is an island. Give verbal sentences:

  1. ¬p
  2. p ∧ ¬q
  3. ¬p ∨ q
  4. p → ¬q
  5. p ↔ q

Solutions:

  1. Jupiter is not a planet
  2. Jupiter is a planet and India is not an island
  3. Jupiter is not a planet or India is an island
  4. If Jupiter is a planet, then India is not an island
  5. Jupiter is a planet if and only if India is an island
Remember: ¬ means "not", ∧ means "and", ∨ means "or", → means "if...then", ↔ means "if and only if"
Exercise 2

Write in symbolic form using p and q:

p: 19 is a prime number
q: All angles of a triangle are equal

  1. 19 is not prime and all angles are equal
  2. 19 is prime or all angles are not equal
  3. 19 is prime and all angles are equal
  4. 19 is not prime

Solutions:

  1. ¬p ∧ q
  2. p ∨ ¬q
  3. p ∧ q
  4. ¬p
Note: "All angles are not equal" is the negation of q (¬q)
Exercise 3

Determine truth values:

  1. If 6+2=5, then milk is white
  2. China is in Europe OR √3 is integer
  3. It is not true that 5+5=9 or Earth is a planet
  4. 11 is a prime number and all sides of a rectangle are equal

Solutions:

  1. True (False → True is True)
  2. False (False OR False is False)
  3. False (Negation of (False OR True) = Negation of True = False
  4. False (True AND False is False)
Remember: A conditional statement is false ONLY when the premise is true and conclusion is false
Exercise 4

Which are propositions?

  1. 4 + 7 = 12
  2. What are you doing?
  3. 3ᵃ ≤ 81, n ∈ ℕ
  4. Peacock is our national bird
  5. How tall this mountain is!

Solutions:

A proposition is a statement that is either true or false.

  1. Proposition (False)
  2. Not proposition (Question)
  3. Not proposition (Depends on n)
  4. Proposition (True in India)
  5. Not proposition (Exclamation)
Exercise 5

Write converse, inverse, and contrapositive:

  1. If x and y are numbers such that x = y, then x² = y²
  2. If a quadrilateral is a square then it is a rectangle

Solutions:

Statement 1:

  • Converse: If x² = y², then x = y
  • Inverse: If x ≠ y, then x² ≠ y²
  • Contrapositive: If x² ≠ y², then x ≠ y

Statement 2:

  • Converse: If a quadrilateral is a rectangle, then it is a square
  • Inverse: If a quadrilateral is not a square, then it is not a rectangle
  • Contrapositive: If a quadrilateral is not a rectangle, then it is not a square
Exercise 6

Construct truth tables:

  1. ¬p ∧ ¬q
  2. ¬(p ∧ ¬q)
  3. (p ∨ q) ∨ ¬q
  4. (¬p → r) ∧ (p ↔ q)

Solution for ¬p ∧ ¬q:

p q ¬p ¬q ¬p ∧ ¬q
T T F F F
T F F T F
F T T F F
F F T T T
Exercise 7

Verify if tautology, contradiction or contingency:

  1. (p ∧ q) ∧ ¬(p ∨ q)
  2. ((p ∨ q) ∧ ¬p) → q
  3. (p → q) ↔ (¬p → q)
  4. ((p → q) ∧ (q → r)) → (p → r)

Solutions:

  1. Contradiction (Always false)
  2. Tautology (Always true)
  3. Contingency (Depends on values)
  4. Tautology (Law of syllogism)
Exercise 8

Show that:

  1. ¬(p ∧ q) ≡ ¬p ∨ ¬q
  2. ¬(p → q) ≡ p ∧ ¬q

Proof for 1 (De Morgan's Law):

p q p ∧ q ¬(p ∧ q) ¬p ¬q ¬p ∨ ¬q
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T

Since ¬(p ∧ q) and ¬p ∨ ¬q have the same truth values, they are equivalent.

Exercise 9

Prove that q → p ≡ ¬p → ¬q

Solution:

We'll prove this using truth tables:

p q q → p ¬p ¬q ¬p → ¬q
T T T F F T
T F T F T F
F T F T F T
F F T T T T

Since q → p and ¬p → ¬q have different truth values (see row 2), they are not equivalent.

Note: This exercise appears to have a mistake in the original problem. The truth table shows they are not equivalent.
Exercise 10

Show that p → q and q → p are not equivalent

Solution:

We can show this with a counterexample:

Let p be True and q be False

p → q: True → False = False

q → p: False → True = True

Since they have different truth values, they are not equivalent.

Remember: The converse of a statement is not equivalent to the original.
Exercise 11

Show that ¬(p → q) ≡ p → ¬q

Solution:

We'll check with a truth table:

p q p → q ¬(p → q) ¬q p → ¬q
T T T F F F
T F F T T T
F T T F F T
F F T F T T

Since ¬(p → q) and p → ¬q have different truth values (rows 3 and 4), they are not equivalent.

Exercise 12

Check if p → (q → p) is a tautology or contradiction without truth table

Solution:

We can rewrite the expression:

p → (q → p) ≡ ¬p ∨ (¬q ∨ p) ≡ (¬p ∨ p) ∨ ¬q ≡ T ∨ ¬q ≡ T

Since the expression simplifies to True regardless of p and q, it is a tautology.

Note: This is always true because if p is true, the conclusion (q→p) is true, and if p is false, the premise is false.
Exercise 13

Check whether ¬(p ∨ q) ∨ (¬p ∧ q) and ¬p are logically equivalent

Solution:

Using truth table:

p q p ∨ q ¬(p ∨ q) ¬p ¬p ∧ q ¬(p ∨ q) ∨ (¬p ∧ q) ¬p
T T T F F F F F
T F T F F F F F
F T T F T T T T
F F F T T F T T

Since both expressions have the same truth values, they are logically equivalent.

Exercise 14

Prove p → (q → r) ≡ (p ∧ q) → r without truth table

Solution:

We can prove this using logical equivalences:

Left side: p → (q → r) ≡ p → (¬q ∨ r) ≡ ¬p ∨ (¬q ∨ r)

Right side: (p ∧ q) → r ≡ ¬(p ∧ q) ∨ r ≡ (¬p ∨ ¬q) ∨ r

Since ¬p ∨ (¬q ∨ r) ≡ (¬p ∨ ¬q) ∨ r (by associative property)

Both sides are equivalent.

Exercise 15

Prove that p → (¬q ∨ r) ≡ ¬p ∨ (¬q ∨ r) using truth table

Solution:

Using truth table:

p q r ¬q ¬q ∨ r p → (¬q ∨ r) ¬p ¬p ∨ (¬q ∨ r)
T T T F T T F T
T T F F F F F F
T F T T T T F T
T F F T T T F T
F T T F T T T T
F T F F F T T T
F F T T T T T T
F F F T T T T T

Since both expressions have identical truth values, they are equivalent.

Truth Table Explorer

Understanding Truth Tables

Truth tables show all possible combinations of truth values for propositions. They help us understand how logical operators work.

Basic truth values: T (True) and F (False)

Basic Truth Tables

Negation (¬p)

p ¬p
T F
F T

Conjunction (p ∧ q)

p q p ∧ q
T T T
T F F
F T F
F F F

Implication (p → q)

p q p → q
T T T
T F F
F T T
F F T
Logical Equivalence Magic

De Morgan's Laws:

¬(p ∧ q) ≡ ¬p ∨ ¬q

¬(p ∨ q) ≡ ¬p ∧ ¬q

Implication equivalence: p → q ≡ ¬p ∨ q

These equivalences help simplify complex logical expressions!

Conditional Statements & Proofs

Converse, Inverse, and Contrapositive

For a conditional statement: p → q

  • Converse: q → p
  • Inverse: ¬p → ¬q
  • Contrapositive: ¬q → ¬p

The contrapositive is logically equivalent to the original statement!

Example: If a quadrilateral is a square, then it is a rectangle

Statement Form Truth Value
Original Square → Rectangle True
Converse Rectangle → Square False
Inverse ¬Square → ¬Rectangle False
Contrapositive ¬Rectangle → ¬Square True