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  1. GATE CS
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  3. Discrete Mathematics

Which of the following is the negation of [forall x, exists y, (alpha(y, z_n) v ~beta(y, z_n))]? A. forall x, exists y, (~alpha(y, z_n) ^ beta(y, z_n)) B. exists x, exists y, (~alpha(y, z_n) ^ beta(y, z_n)) C. forall x, forall y, (~alpha(y, z_n) v beta(y, z_n)) D. exists x, forall y, (~alpha(y, z_n) ^ beta(y, z_n))

GATE 2008 · Discrete Mathematics · First Order Logic · medium

Answer: D. exists x, forall y, (~alpha(y, z_n) ^ beta(y, z_n))

  1. Negate the outer universal quantifier: ~[forall x, exists y, (alpha(y,z_n) v ~beta(y,z_n))] = exists x, ~[exists y, (alpha(y,z_n) v ~beta(y,z_n))]
  2. Negate the inner existential, then apply De Morgan: exists x, ~[exists y, (alpha v ~beta)] = exists x, forall y, ~(alpha(y,z_n) v ~beta(y,z_n)) = exists x, forall y, (~alpha(y,z_n) ^ ~~beta(y,z_n)) = exists x, forall y, (~alpha(y,z_n) ^ beta(y,z_n))