Check whether the statement p → (q → p) is a tautology or a contradiction without using the truth table

Sum

P → (q → p)

≡ P → (¬q v p) ......[∵ implication law]

≡ ¬p v (¬q v p) ......[∵ implication law]

≡ ¬p v (p v ¬q) .......[∵ Commutative law]

≡ (¬p V p) v (¬p v ¬q) ......[∵ Distribution law]

≡ T v ¬(p ∧ q) ≡ T ......[Tautology]

Hence p → (q → p) is a Tautology.

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Mathematical Logic

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Chapter 12: Discrete Mathematics - Exercise 12.2 [Page 249]

Chapter 12 Discrete Mathematics
Exercise 12.2 | Q 12 | Page 249

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

¬ P

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

p → ¬q

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

p ↔ q

Which one of the following sentences is a proposition?

What are you doing?

Write the converse, inverse, and contrapositive of the following implication.

If x and y are numbers such that x = y, then x² = y²

Construct the truth table for the following statement

(¬p → r) ∧ (p ↔ q)

Verify whether the following compound propositions are tautologies or contradictions or contingency.

(p ∧ q) ∧¬ (p v q)

Verify whether the following compound propositions are tautologies or contradictions or contingency.

((p → q) ∧ (q → r)) → (p → r)

Prove that q → p ≡ ¬p → ¬q

Show that ¬(p ↔ q) ≡ p ↔ ¬q