Determine, using logical equivalence laws, if(¬(A ∧ ¬B)) → (A → B)is a tautology; i.e. that it can be reduced to T. In each step name the equivalence law you are using

Step 1: Apply De Morgan's Law The first step is to apply De Morgan's Law to the expression ¬(A ∧ ¬B). De Morgan's Law states that the negation of a conjunction is equivalent to the disjunction of the negations. Therefore, ¬(A ∧ ¬B) is equivalent to ¬A ∨ B.

So, the expression becomes (¬A ∨ B) → (A → B).

Step 2: Apply Material Implication Next, apply the law of Material Implication to both implications in the expression. Material Implication states that (P → Q) is equivalent to (¬P ∨ Q). Therefore, (¬A ∨ B) → (A → B) is equivalent to ¬(¬A ∨ B) ∨ (¬A ∨ B), and (A → B) is equivalent to (¬A ∨ B).

So, the expression becomes ¬(¬A ∨ B) ∨ (¬A ∨ B).

Step 3: Apply Double Negation Then, apply the law of Double Negation to ¬(¬A). Double Negation states that ¬(¬P) is equivalent to P. Therefore, ¬(¬A) is equivalent to A.

So, the expression becomes ¬A ∨ (¬A ∨ B).

Step 4: Apply Associative Law Next, apply the Associative Law to the expression. The Associative Law states that (P ∨ (Q ∨ R)) is equivalent to ((P ∨ Q) ∨ R). Therefore, ¬A ∨ (¬A ∨ B) is equivalent to (¬A ∨ ¬A) ∨ B.

So, the expression becomes (¬A ∨ ¬A) ∨ B.

Step 5: Apply Idempotent Law Finally, apply the Idempotent Law to the expression. The Idempotent Law states that (P ∨ P) is equivalent to P. Therefore, (¬A ∨ ¬A) is equivalent to ¬A.

So, the expression becomes ¬A ∨ B.

Since the expression has been reduced to a disjunction of two literals, it is a tautology. Therefore, the original expression (¬(A ∧ ¬B)) → (A → B) is a tautology.