Let a(x, y), b(x, y,) and c(x, y) be three statements with variables x and y chosen from some universe. Consider the following statement:
(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)]
Which one of the following is its equivalent?
(A) (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]
(B) (∃x)(∀y)[(a(x, y) ∨ b(x, y)) ∧¬ c(x, y)]
(C) ¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)]
(D) ¬ (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]
Answer: (C)
Explanation: Choice (C) is
= ¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)]
= ¬ (∀x)(∃y)[ a∧ b → c ]
= ¬ (∀x)(∃y)[ (ab)’ + c]
= ∃ x ∀ y[ (ab)’ + c]’
= ∃ x ∀ y[ abc’ ]
= ∃ x ∀ y[ a ∧ b ∧ c ¬c]
Which is same as the given expression.
(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)]
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