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Are these two logical formulas logically equivalent or not? What I am asking is whether these two formulas mean the same thing or not.

(1) ∃x ∀y ((A(x) ∧ B(y)) → ¬C(x, y) )

(2) ∃x (A(x) ∧ ∀y (B(y) → ¬C(x, y) ))

Many thanks in advance

daveid
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  • Does $A(x)\land B(y)\rightarrow\neg C(x,y)$ mean $(A(x)\land B(y))\rightarrow\neg C(x,y)$ or does it mean $A(x)\land(B(y)\rightarrow\neg C(x,y))$? – bof Jan 11 '18 at 23:34
  • (1) means (A(x) ∧ B(y)) → ¬C(x,y), I'll edit it to clarify. – daveid Jan 11 '18 at 23:40
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    If $A(x)$ is false for all $x,$ then (1) is true but (2) is false. So the two formulas are not equivalent. – bof Jan 12 '18 at 00:01

1 Answers1

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Consider a universe where $A(x)$ never holds. What are the truth values of the two statements?

BallBoy
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