Question

The variable p is true, q is false, and the truth value for variable r is unknown. Indicate whether the truth value of each logical expression is true, false, or unknown.

A) p→(q∧r)
B) (p∧q)→r

Answer 1

Expression A is false as it involves a true premise leading to a false conclusion. Expression B is true because it involves a false premise, which makes the implication true regardless of the conclusion's truth value.

Step-by-step explanation:

The question involves determining the truth value of given logical expressions based on the provided truth values for the variables p, q, and r.

Logical Expression Analysis

• A) p→(q∧r): Since p is true and q is false, the expression q∧r is false regardless of the value of r. In logic, if the premise is true and the conclusion is false, the implication is false. Therefore, the truth value for this expression is false.
• B) (p∧q)→r: The expression p∧q is false because q is false and a conjunction is only true if both parts are true. In an implication, if the premise is false, the whole implication is considered true. Therefore, the truth value for this expression is true.

Answer 2

A) The truth value for the logical expression p→(q∧r) is unknown.

B) The truth value for the logical expression (p∧q)→r is false.

Step-by-step explanation:

A) For the expression p→(q∧r), the implication involves q∧r. Since q is false, the entire expression depends on the truth value of r, which is unknown. Therefore, the truth value for the entire expression is also unknown.

B) In the case of (p∧q)→r, the antecedent (p∧q) requires both p and q to be true. However, q is false in this scenario. Therefore, the antecedent is false, making the entire implication true regardless of the truth value of r.

In logic, unknown truth values are often treated separately from true or false values. In this case, the presence of an unknown truth value makes the truth value of the entire expression uncertain. In contrast, the false antecedent in the second expression (B) makes the implication false, regardless of the unknown truth value of r. It illustrates how logical expressions can be influenced by the truth values of their constituent variables.