Using a truth table, prove that $ p \Rightarrow q $ is equivalent to $ \sim q \Rightarrow \sim p $.

Question

Accepted Answer

To prove this, a truth table is constructed to evaluate both propositions across all possible combinations of truth values for variables $ p $ and $ q $.

    Truth Table for Equivalence Proof:
   
        
            
                $ p $
                $ q $
                $ \sim q $
                $ \sim p $
                $ p \Rightarrow q $
                $ \sim q \Rightarrow \sim p $
            
        
        
            
                0
                0
                1
                1
                1
                1
            
            
                0
                1
                0
                1
                1
                1
            
            
                1
                0
                1
                0
                0
                0
            
            
                1
                1
                0
                0
                1
                1
            
        
    

    Conclusion:
    From the truth table above, it is evident that the columns for $ p \Rightarrow q $ and $ \sim q \Rightarrow \sim p $ are identical, meaning they possess the same truth set: (1, 1, 0, 1).
    Hence, it is proved that $ p \Rightarrow q \equiv \sim q \Rightarrow \sim p $. This logical rule is also formally known as transposition.

\( p \)	\( q \)	\( \sim q \)	\( \sim p \)	\( p \Rightarrow q \)	\( \sim q \Rightarrow \sim p \)
0	0	1	1	1	1
0	1	0	1	1	1
1	0	1	0	0	0
1	1	0	0	1	1

Using a truth table, prove that \( p \Rightarrow q \) is equivalent to \( \sim q \Rightarrow \sim p \).

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