please do all three and bonus? Prove or Disprove: It A. B, C are sets with nonem

Question

please do all three and bonus?

Prove or Disprove: It A. B, C are sets with nonempty intersections AnB, AnC. and BnC, the three-way intersection AnBnC must also he nonempty. Give proofs of parts (3)-(6) of the inverse image theorem for functions, i.e. the section titled Properties of Pre-Image Sets on page 5 of the Sets, Relations and Functions lecture posted on the class website. Also, for each of these four parts, determine explicitly what happens in the case where f(x) = c (V x epsilon X) is a constant function. Prove every in Unite set X contains an infinite subset A c X. where A can be placed in a one-to-one correspondence with N = {1,2,3,...}.(Bonus: Extend this argument a bit to prove every infinite set has the property of being in a one-to-one correspondence with a proper subset of itself.)

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Chegg is not allowing me to write long answers, so I will post the answer as a comment to this post.

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