Prove that $ p \Leftrightarrow q = q \Leftrightarrow p $.

Question

Accepted Answer

To prove this equivalence, a truth table is constructed to evaluate the truth values for both $ p \Leftrightarrow q $ and $ q \Leftrightarrow p $ across all possible combinations of $ p $ and $ q $.

    Truth Table for Bi-conditional Equivalence:
    
        
            
                $ p $
                $ q $
                $ p \Leftrightarrow q $
                $ q \Leftrightarrow p $
            
        
        
            
                0
                0
                1
                1
            
            
                0
                1
                0
                0
            
            
                1
                0
                0
                0
            
            
                1
                1
                1
                1
            
        
    

    Conclusion:
    From the truth table above, it is found that both propositions $ p \Leftrightarrow q $ and $ q \Leftrightarrow p $ possess the same truth set: (1, 0, 0, 1).
    Hence, it is proved that: $ p \Leftrightarrow q = q \Leftrightarrow p $.

\( p \)	\( q \)	\( p \Leftrightarrow q \)	\( q \Leftrightarrow p \)
0	0	1	1
0	1	0	0
1	0	0	0
1	1	1	1

Prove that \( p \Leftrightarrow q = q \Leftrightarrow p \).

Answer with Explanation