Question

The number of values of $r \in\\{p, q, \sim p, \sim q\\}$ for which $((p \wedge q) \Rightarrow(r \vee q)) \wedge((p \wedge r) \Rightarrow q)$ is a tautology, is :

• A. 1
• B. 2
• C. 4
• D. 3

Answer 1

Answer: (B)

2

Answer 2

((p∧q)⇒(r∨q))∧((p∧r)⇒q)
We know, p⇒q is equivalent to
∼p∨q
(∼(p∧q)∨(r∨q))∧(∼(p∧r))∨q))
⇒(∼p∨∼q∨r∨q)∧(∼p∨∼r∨q)
⇒(∼p∨r∨t)∧(∼p∨∼r∨q)
⇒(t)∧(∼p∨∼r∨q)
For this to be tautology, (∼p∨∼r∨q) must be always true which follows for r=∼p or r=q.
So, the correct option is (B) : 2