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

We verify the equivalence of \( B \Rightarrow ((\sim A) \lor B) \) by using truth tables. 
Step 1: Analyze the given expression The statement is \( B \Rightarrow ((\sim A) \lor B) \). By the definition of implication: \[ P \Rightarrow Q \equiv (\sim P) \lor Q, \] the statement can be rewritten as: \[ (\sim B) \lor ((\sim A) \lor B). \] 
Step 2: Construct the truth table for \( B \Rightarrow ((\sim A) \lor B) \) 

Step 3: Verify equivalence with the options 
(A) Option (1): \( B \Rightarrow (A \Rightarrow B) \) Simplify \( A \Rightarrow B \equiv (\sim A) \lor B \). Thus: \[ B \Rightarrow ((\sim A) \lor B). \] The truth table matches exactly with \( B \Rightarrow ((\sim A) \lor B) \). 

(B)Option (3): \( A \Rightarrow ((\sim A) \Rightarrow B) \) Simplify \( (\sim A) \Rightarrow B \equiv A \lor B \). Thus: \[ A \Rightarrow ((\sim A) \lor B) \equiv (\sim A) \lor ((\sim A) \lor B), \] which matches the truth table of \( B \Rightarrow ((\sim A) \lor B) \). 

(C)Option (4): \( B \Rightarrow ((\sim A) \Rightarrow B) \) Simplify \( (\sim A) \Rightarrow B \equiv A \lor B \). Thus: \[ B \Rightarrow (A \lor B), \] which also matches the truth table of \( B \Rightarrow ((\sim A) \lor B) \).