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I am studying discrete math at university. However, the materials don't seem to explain non-numeric predicate statements in much detail. I am asked to prove that a predicate statement is false using nested quantifiers and give reasons. The statement is this:

For all the simple things you have done to me, there exists one thing that makes me happy.


My predicate statement is something like this but I'm confused about how to prove it is false.

Let
UD= All things
S(x)= x is a simple thing you have done to me
H(x)= x is a thing that makes me happy

∀x ( S(x) → ∃!y H(y) )

If someone could please explain how you are supposed to go about this, or perhaps provide a link, I would be very grateful.

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1 Answer 1

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For all the simple things you have done to me, there exists one thing that makes me happy.

(We presume that the things that make them happy is a subset of the aforementioned simple things.)

  1. Your translation $$∀x ( S(x) → ∃!y H(y) )$$ is incorrect because “there exists one thing” doesn't mean “there exists exactly one thing”.

  2. The given statement is literally translated as $$∀x∃x ( S(x) → H(x) ),$$ which is logically equivalent to $$∀x ( S(x) → H(x) ),$$ which can be translated back as “Every simple thing that you have done to me makes me happy.”

The speaker probably means “Among the simple things you have done to me, there exists one thing that makes me happy” instead.

In this case, the translation is $$∃x S(x) → ∃x ( H(x) ∧ S(x) ),$$ which is logically equivalent to $$∀x ∃y (S(x) → ( H(y) ∧ S(y) )$$ and to $$∃y ∀x (S(x) → ( H(y) ∧ S(y) ).$$

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