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 relations R,S, and T

T
o (S o R) = (T o S) o R.
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(3) Prove: If f and g are one-to-one functions, g o f is also a one-to-one function, and (g o f)1
= f1o g1.
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(4) Let f be a function A onto B. Define a relation E in A by: aEb if and only if f(a) = f(b).

(a) Show that E is an equivalence relation on A.
(b) Define a function φ on A/E onto B by φ([a]
E) = f(a) (verify that φ([a]E)=φ([a']E) if [a]E = [a']E).
(c) Let j be the function on A onto A/E given by j(a)=[a]E. Show that φ o j = f.