Question

Proposition 2B Given the following proposition: [(A⊃Y)≡(B⊃∼X)]∨∼[(B⋅∼X)≡(Y⋅A)] Given that A and B are true and X and Y are false, determine the truth value of Proposition 2 B.

a.True.
b. False

Answer 1

Upon substituting true for A and B, and false for X and Y in the given proposition and evaluating, we find that the resulting disjunction is false, leading to the conclusion that Proposition 2B is false.

Step-by-step explanation:

To determine the truth value of Proposition 2B, which is [(A⊃Y)≡(B⊃∼ X)]∨∼['(B·∼ X)≡(Y·A)], we need to substitute the given values for A, B, X, and Y into the proposition and evaluate it.

Let's start by substituting the given values where A and B are true (T) and X and Y are false (F):

• [(T⊃F)≡(T⊃T)]∨∼['(T·T)≡(F·T)]

Now let's evaluate each part of the proposition:

• The implication (A ⊃ Y) is false because true does not imply false.
• The implication (B ⊃ ∼ X) is true because true implies not false, which is true.
• The equivalence [(A⊃Y)≡(B⊃∼ X)] is false because one side is false and the other is true.
• The conjunction (B · ∼ X) is true because both true and not false are true.
• The conjunction (Y · A) is false because false and true results in false.
• The equivalence [(B·∼ X)≡(Y·A)] is true because both sides are equivalent (both false).

The negation of the equivalence, ∼['(B·∼ X)≡(Y·A)'], is therefore false.

Finally, the disjunctive syllogism tells us that a disjunction is true if at least one of the parts is true. Since both parts of the disjunction are false, the entire proposition is also false.

Therefore, the answer to the question is: b. False.