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Answer:
d.(p ∧ ¬p) .
Step-by-step explanation:
A tautology in logic is a formula or assertion that is true in every possible interpretation. This means that the statement is always true, regardless of the truth values of its variables.
Let's evaluate each of the given statements:
a.) ¬(p ∧ q) → ¬(p ∨ q)
b.) (p ∧ ¬p) ↔ (q ∨ ¬q)
c.) (p ∧ ¬p) → ¬(q ∨ ¬q)
d.) (p ↔ ¬p) ∧ (q ∨ ¬q)
Evaluation
a.) ¬(p ∧ q) → ¬(p ∨ q)
This is not a tautology. There are cases where this statement can be false. For example, if p is true and q is false, then ¬(p ∧ q) is true, but ¬(p ∨ q) is false.
b.) (p ∧ ¬p) ↔ (q ∨ ¬q)
This is a tautology. The left side (p ∧ ¬p) is a contradiction and is always false. The right side (q ∨ ¬q) is a tautology and is always true. A false statement is equivalent to a true statement in logic, so this entire statement is always true.
c.) (p ∧ ¬p) → ¬(q ∨ ¬q)
This is not a tautology. The left side (p ∧ ¬p) is a contradiction and is always false. The right side ¬(q ∨ ¬q) is a contradiction and is always false. A false statement implies a false statement is not always true.
d.) (p ↔ ¬p) ∧ (q ∨ ¬q)
This is not a tautology. The left side (p ↔ ¬p) is a contradiction and is always false. The right side (q ∨ ¬q) is a tautology and is always true. A false statement and a true statement is not always true.