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I'm working through my logic courses text book questions, which sadly has no answers provided.

And I'm stuck on this question that asks what the best symbolization of this sentence out of the options would be.

The sentence is:

Everybody loves somebody that loves them.

And the four options are:

  1. ∀x ∃y ( L(x,y) ∧ L(y,x) )
  2. ∀x ∀y ( L(x,y) -> L(y,x) )
  3. ∃y ∀x ( L(x,y) ∧ L(y,x) )
  4. ∃x ∃y ( L(x,y) ∧ L(y,x) )

This is my current idea

1 would give: For everyone, there is someone that they love and are loved back by them

2 would give: If anyone loves anyone then that person loves them back

3 would give: There is someone, that everyone loves and everyone loves them back

  1. there is someone, and they love someone that loves them back

And so with that I would guess that the first one is the best.

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Yes, (1) is correct since it's discussing "for all people, there exists someone, who loves them and they love back"

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