Prove that $ X + X' $ is a tautology and $ X \cdot X' $ is a contradiction.

Question

Accepted Answer

To prove these, a truth table is constructed to evaluate both expressions for all possible values of the variable $ X $ and its complement $ X' $.

    Truth Table for Complementary Expressions:
   
        
            
                $ X $
                $ X' $
                $ X + X' $
                $ X \cdot X' $
            
        
        
            
                0
                1
                1
                0
            
            
                1
                0
                1
                0
            
        
    

    Conclusion:
    
        $ X + X' $ has all 1's in its truth set, hence it is a tautology.
        $ X \cdot X' $ has all 0's in its truth set, hence it is a contradiction.

\( X \)	\( X' \)	\( X + X' \)	\( X \cdot X' \)
0	1	1	0
1	0	1	0

Prove that \( X + X' \) is a tautology and \( X \cdot X' \) is a contradiction.

Answer with Explanation