The statement A → (B → A) is equivalent to :
Question:

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

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

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

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

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


Correct Option: , 4

Solution:

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

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

$\equiv \sim \mathrm{A} \vee(\sim \mathrm{B} \vee \mathrm{A})$

$\equiv(\sim \mathrm{A} \vee \mathrm{A}) \vee \sim \mathrm{B}$

$\equiv \mathrm{T} \vee \sim \mathrm{B} \equiv \mathrm{T}$

$\therefore \mathrm{T} \vee \mathrm{B}=\mathrm{T}$

$\equiv(\sim \mathrm{A} \vee \mathrm{A}) \vee \mathrm{B}$

$\equiv \sim \mathrm{A} \vee(\mathrm{A} \vee \mathrm{B})$

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

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