The statement
Question:

The statement $\mathrm{A} \rightarrow(\mathrm{B} \rightarrow \mathrm{A})$ is equivalent to:

  1. (1) $\mathrm{A} \rightarrow(\mathrm{A} \wedge \mathrm{B})$

  2. (2) $\mathrm{A} \rightarrow(\mathrm{A} \vee \mathrm{B})$

  3. (3) $\mathrm{A} \rightarrow(\mathrm{A} \rightarrow \mathrm{B})$

  4. (4) $\mathrm{A} \rightarrow(\mathrm{A} \leftrightarrow \mathrm{B})$


Correct Option: , 2

Solution:

$A \rightarrow(B \rightarrow A)$

$\Rightarrow \mathrm{A} \rightarrow(\sim \mathrm{B} \vee \mathrm{A})$

$\Rightarrow \sim A \vee(\sim B \vee A)$

$\Rightarrow \sim B \vee(\sim A \vee A)$

$\Rightarrow \sim B \vee t$

$=\mathrm{t}$ (tautology) From options:

(2) $\mathrm{A} \rightarrow(\mathrm{A} \vee \mathrm{B})$

$\Rightarrow \sim A \vee(A \vee B)$

$\Rightarrow(\sim A \vee A) \vee B$

$\Rightarrow t \vee B$

$\Rightarrow t$

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