Question

Which of the following is FALSE? a. (x→(x∨y))

b. ((∼x→y)^(∼x^∼y))→y
c. ((x→y)^x)→y
d. ((x∨y)→(∼x∨∼y))

Answer 1

The false statement is ((x→y)^x)→y.

Step-by-step explanation:

The correct answer is c. ((x→y)^x)→y.

To determine which option is false, we can evaluate each of them:

1. a. (x→(x∨y))
2. This statement is true because if x is true, then x∨y is also true, and the implication holds.
3. b. ((∼x→y)^(∼x^∼y))→y
4. This statement is true. If (∼x→y)^(∼x^∼y) is true, then ∼x→y is true, and since y is the conclusion, the statement holds.
5. c. ((x→y)^x)→y
6. This statement is false. If x is true and x→y is true, it does not necessarily mean that y is true. The conclusion is false, so the statement is false.
7. d. ((x∨y)→(∼x∨∼y))
8. This statement is true. If x∨y is true, then (∼x∨∼y) is also true, and the implication holds.

Therefore, option c is the false statement. ((x→y)^x)→y is not always true.