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Question 7

The statement among the following that is a tautology is:

We need to identify which statement is a tautology (always true regardless of truth values of $$A$$ and $$B$$).

Option D is $$(A \wedge (A \to B)) \to B$$. This is the well-known rule of Modus Ponens.

Let us verify using a truth table. When $$A$$ is True and $$B$$ is True: $$A \to B$$ is True, so $$A \wedge (A \to B)$$ is True, and $$\text{True} \to \text{True}$$ is True.

When $$A$$ is True and $$B$$ is False: $$A \to B$$ is False, so $$A \wedge (A \to B)$$ is False, and $$\text{False} \to \text{False}$$ is True.

When $$A$$ is False and $$B$$ is True: $$A \to B$$ is True, so $$A \wedge (A \to B)$$ is False, and $$\text{False} \to \text{True}$$ is True.

When $$A$$ is False and $$B$$ is False: $$A \to B$$ is True, so $$A \wedge (A \to B)$$ is False, and $$\text{False} \to \text{False}$$ is True.

In all cases the result is True, confirming this is a tautology.

Hence, the correct answer is Option D.

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