Question

The statement (p∧ (-q)) v ((~ p) ∧ q) v ((~q)) is equivalent to

• A. (~ p) v (~q)
• B. p ∨(~ q)
• C. (~p) ∨ q
• D. p ∨ q

Answer 1

The Correct Option is A

To find the equivalent statement of the given expression, let's use the properties of logic statements involving logical conjunctions (∧), disjunctions (∨), and negations (~).

1. The given statement is: \((p \land \neg q) \lor (\neg p \land q) \lor \neg q\). 
2. Let's consider distributing and simplify each part:
   • The term \((p \land \neg q) \lor (\neg p \land q)\) appears to be like an XOR operation (exclusive OR), which means it is true when exactly one of the variables is true.
   • In a simpler form, \((p \land \neg q) \lor (\neg p \land q)\) can cover all possibilities unless both are false or both are true (we simplify directly for logical equivalence purposes).
3. Now, with the third part \(\neg q\), we have:
   • If \(\neg q\) is true, the whole statement is true regardless of \(p\) as any \((False \lor True = True)\).
4. Let's evaluate the combined expression.
   • Note: Simplifying through logical inference: If \(\neg q\) is explicitly included, then whenever \(q\) is false, the entire expression satisfies as true.
5. This covers cases when \(\neg q\) is true. But observe:
   • For any term that starts with \((\neg p \land q)\), \(\neg q\) is always present.
6. Revisiting the expression, since we have to check for variable impact:
   • \((p \land \neg q) \lor (\neg p \land q) \lor \neg q = \neg p \lor \neg q\) as their presence will satisfy the conditions resulting in simplification into disjoint expressions \((\lor)\).
7. Thus, the statement is logically equivalent to: \(\neg p \lor \neg q\).

Therefore, the correct answer is the option

(~ p) v (~q)

.