Question

The statement $A\to (B\to A)$ is equivalent to:

• A. $A\to (B\wedge A)$
• B. $A\to (B\vee A)$
• C. $A\to (B\to A)$
• D. $A\to (B\leftrightarrow A)$

Answer 1

The correct option is B

$A\to (B\vee A)$

Evaluate the equivalent value of the given condition

Given, $A\to (B\to A)$

We can solve this as,

$$\begin{gathered} \equiv A\to (B\to A) \\ \equiv A\to (~B\vee A) \\ \equiv ~A\vee (~B\vee A) \\ \equiv (~A\vee A)\vee ~B \end{gathered}$$

Let, $(~A\vee A)=T$

We know that if $T\vee ~B=T$, so, $T\vee B\equiv T$

So,

$$\begin{gathered} \equiv (~A\vee A)\vee B \\ \equiv ~A\vee (A\vee B) \\ \equiv A\to (A\vee B) \end{gathered}$$

Therefore, option (B), $A\to (A\vee B)$ is the correct answer.