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Mathematics Question on mathematical reasoning

Which of the following statements is a tautology?

A

∼(p ∧ ∼ q)→p ∨ q\sim\left(p\, \wedge\, \sim\, q\right) \rightarrow p\, \vee\, q

B

∼(p ∨ ∼ q)→ p ∧q\sim\left(p\, \vee\, \sim\, q\right) \rightarrow\, p\, \wedge q

C

p∨(∼q)→p∧qp \vee \left(\sim q\right) \rightarrow p \wedge q

D

∼(p ∨ ∼q)→p ∨ q\sim\left(p\, \vee\, \sim q\right) \rightarrow p\, \vee\, q

Answer

∼(p ∨ ∼q)→p ∨ q\sim\left(p\, \vee\, \sim q\right) \rightarrow p\, \vee\, q

Explanation

Solution

∼(p ∨ ∼q)→p ∨ q\sim\left(p\, \vee\, \sim q\right) \rightarrow p\, \vee\, q
(∼p∧q)→p∨q\left(\sim p \wedge q\right) \to p \vee q
\sim\left\\{\left(\sim p \wedge q\right)\wedge \left(\sim p \wedge \sim q\right)\right\\}
∼(∼p∧f)\sim\left(\sim p \wedge f\right)

Which of the following statements is a tautology
So, the correct option is (D): ∼(p ∨ ∼q)→p ∨ q\sim\left(p\, \vee\, \sim q\right) \rightarrow p\, \vee\, q.