Need help with 2(d) and 3(a). 2. Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9} and T = {2, 4, 6,8}. Let R be the relation on P (S), the power set of S, defined by:For all A, B EP (S), (A, B) = R if and only if AUT=BUT.(a) Prove that R is an equivalence relation on P (S).(b) How many equivalence classes are there? Explain.(c) How many elements does [{1, 2}], the equivalence class of {1,2}, have? Explain.(d) How many subsets X € P (S) are there so that |X| = 5 and X is not an element of [{1,2}]? Explain. Questions:1. Let A = {1, 2, 3, 4, 5). Let F be the set of all functions from A to A. Let R be the relation on F defined by: for allf, g = F, fRg if and only if ƒ (1) = g (2) or f (2) = g(1). The identity function IA: A → A is defined by IA (x) = x forevery x EA.(a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers.(b) Is it true that for all functions ƒ € F, there is a function g so that fRg? Prove your answer.(c) Is it true that for all functions ƒ € F, there a function g so that (f, g) & R? Prove your answer.(d) How many functions f: A→ A are there so that fRIA? Explain.3. Let A = {1,2,3,4,5}. Let f : A → A be the function defined by f={(1,2), (2, 2), (3, 1), (4,1), (5,5)}. Next,we define the function g : P (A) → P (A) by putting g(Ø) = Ø and for non-empty subset V of A, we put g (V) ={f(x) | x € V}.Let R be the relation on P (A) defined by:For any X, Y EP (A), (X, Y) = R if and only if g (X) = g(Y).(a) Prove that R is an equivalence relation on P (A).(b) How many equivalence classes are there? Explain.(c) Find three different elements of [{1,3,5}], the equivalence class of {1,3,5}.(d) How many elements does [{1,3,5}] have? Explain.:

Kindly refer to Step 2 for the solution.

2. Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9} and T = {2,4,6,8}. Let R be the relation on P (S), the power set of S, defined by: For all A, B EP (S), (A, B) € R if and only if AUT=BUT. (a) Prove that R is an equivalence relation on P (S). (b) How many equivalence classes are there? Explain. (c) How many elements does [{1, 2}], the equivalence class of {1,2}, have? Explain. (d) How many subsets XP (S) are there so that |X| = 5 and X is not an element of [{1, 2}]? Explain.