Question

The statement B⇒((∼A) ∨ B) is equivalent to :

• A. B⇒(A⇒B)
• B. A⇒((∼A)⇒B)
• C. A⇒(A⇒B)
• D. B⇒((∼A)⇒B)

Answer 1

The Correct Option is A

To solve the problem, we need to find an equivalent expression for the statement \( B \Rightarrow ((\sim A) \lor B) \).

The logical implication \( P \Rightarrow Q \) can be rewritten using logical operations as \( \sim P \lor Q \). Therefore,

B \Rightarrow ((\sim A) \lor B) becomes \sim B \lor ((\sim A) \lor B).

We can simplify the expression \(\sim B \lor ((\sim A) \lor B)\) using the associative and commutative laws of logic:

• Using the associative law: \(\sim B \lor ((\sim A) \lor B) = (\sim B \lor \sim A) \lor B\).
• Using the law of redundancy (also known as absorption law), \((P \lor Q) \lor P\) simplifies to \(P \lor Q\).

Applying absorption law:

• \((\sim B \lor \sim A) \lor B\) reduces to \(B \lor \sim A\), since \((\sim A \lor B) \lor B = \sim A \lor B\).

So, the original statement \( B \Rightarrow ((\sim A) \lor B) \) simplifies to \( B \lor \sim A \).

We aim to match this with the options:

• Option 1: \(B \Rightarrow (A \Rightarrow B)\)

For \( A \Rightarrow B \) we have \(\sim A \lor B\). So \( B \Rightarrow (A \Rightarrow B)\) translates to:

• \(B \Rightarrow (\sim A \lor B)\) which gives us \(\sim B \lor (\sim A \lor B)\), and this simplifies again using the same steps to \( B \lor \sim A \).

This matches exactly what we derived: \( B \lor \sim A \).

Thus, the statement \( B \Rightarrow ((\sim A) \lor B) \) is equivalent to \( B \Rightarrow (A \Rightarrow B) \).

Correct answer: \( B \Rightarrow (A \Rightarrow B) \).