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

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Answer with Explanation

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.