Prove that $ p \Rightarrow q = \bar{p} + q $.

Question

Accepted Answer

To prove this equivalence, a truth table is constructed to compare the truth values of the conditional expression $ p \Rightarrow q $ and the expression $ \bar{p} + q $ for all possible combinations of $ p $ and $ q $.

    Truth Table for Identity Proof:
    
        
            
                $ p $
                $ q $
                $ \bar{p} $
                $ p \Rightarrow q $
                $ \bar{p} + q $
            
        
        
            
                0
                0
                1
                1
                1
            
            
                0
                1
                1
                1
                1
            
            
                1
                0
                0
                0
                0
            
            
                1
                1
                0
                1
                1
            
        
    

    Conclusion:
    From the truth table above, it is observed that the columns for $ p \Rightarrow q $ and $ \bar{p} + q $ possess the same truth set: (1, 1, 0, 1).
    Therefore, it is proved that: $ p \Rightarrow q = \bar{p} + q $.

\( p \)	\( q \)	\( \bar{p} \)	\( p \Rightarrow q \)	\( \bar{p} + q \)
0	0	1	1	1
0	1	1	1	1
1	0	0	0	0
1	1	0	1	1

Prove that \( p \Rightarrow q = \bar{p} + q \).

Answer with Explanation