Question

Consider the following statements.
\(p\): If \(3^4 > 4^3\), then \(3^3 > 4^4\)
\(q\): The roots of the equation \(x^2-2x+2 = 0\) are real if and only if Mumbai is in Maharashtra.
\(r\): Statement \(p\) is true or statement \(q\) is false.
Which of the following has truth value T (true)?

• A. \((p∨q)∧r\)
• B. \(p∨(q∧r)\)
• C. \(p∧(q∨r)\)
• D. \((p∧q)∨r\)

Hint:

Find the truth values of p, q and r first, then evaluate each option.

Answer 1

The Correct Option is D

Step 1: Truth value of p:
\(3^4 = 81 > 4^3 = 64\) is true. \(3^3 = 27 > 4^4 = 256\) is false. A true hypothesis with a false conclusion makes "if...then" false. So \(p\) is F.

Step 2: Truth value of q:
The roots of \(x^2 - 2x + 2 = 0\) have discriminant \(4 - 8 = -4 < 0\), so the roots are not real: statement F. "Mumbai is in Maharashtra" is T. A biconditional between F and T is F. So \(q\) is F.

Step 3: Truth value of r:
\(r\) is "\(p\) is true or \(q\) is false" which is \(p \lor \sim q = F \lor T = T\). So \(r\) is T.

Step 4: Evaluate the options:
(A) \((p \lor q) \land r = (F \lor F) \land T = F\).
(B) \(p \lor (q \land r) = F \lor (F \land T) = F\).
(C) \(p \land (q \lor r) = F \land T = F\).
(D) \((p \land q) \lor r = F \lor T = T\).

Final Answer:
Only option (D) is true. \[ \boxed{(p\wedge q)\vee r} \]