Have an assignment due tomorrow and I'm stuck on these 4 questions. I'm fairly new to formal proofs and would greatly appreciate if someone could show me how these questions are done. Thanks

(1) Let X= { ∅, { ∅}}, Y= P(X) [power series of X], Determine:

(a) ∊Y

(b) IdY

Determine the domain, range, and field of both relations.

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(2) Prove that for any three binary relationsR,S,andTo (

TSoR) = (ToS) oR.-------------------------------------------------------------------------

(3) Prove: Iffandgare one-to-one functions,gofis also a one-to-one function, and (gof)–1 =f–1og–1.

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(4) Letfbe a functionAontoB. Define a relationEinAby:aEbif and only if f(a) = f(b).

(a) Show thatEis an equivalence relation on A.

(b) Define a function φ onA/EontoBby φ([a]E) =f(a) (verify that φ([a]E)=φ([a']E) if [a]E= [a']E).

(c) Letjbe the function onAontoA/Egiven byj(a)=[a]E.Show that φ oj=f.