Predicate Logic - What is the difference between these two answers?

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Main Question

Okay so the question at hand is to describe this statement using set theory and possible from logic.

Everybody is friends with everybody

My answer would be for this:

∀x∈S |{z∈S, f(x,x,z)}| = |S|

however could someone clarify if this is the same as saying

∀x∈S |{z∈S, f(x,z,z)}| = |S|

If not, could you please explain the difference .

Thank you,