I have attached the solution to the picture below :
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y . Prove that for every subset A ⊂ X: (a) (10 points) A ⊂ F^(−1) (F(A)). (b) (10 points) F ^(−1) (F(A)) ⊂ A
5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI 5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI
4. Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y. Prove that for every subset A CX: (a) (10 points) AC F-(F(A)). (b) (10 points) F-1(F(A)) C A.
4. Consider a function f : X → Y. 4a) (5 pts) Let C, D be subsets of Y. Prove that f (CND)sf1(C)nf-1(D). 4b) (10 pts) Let A, B be subsets of X and assume the function f be one-to- one. Prove that f(A) n f(B)Cf(An B) (Justify each of your steps.) 4c) (4pts) Find an example showing that if the function f is not one-to-on the inequality (1) is violated.
Problem 6 (6 points) Let f(x) = u(x, y) + iv(x, y) be a analytic function on D and extends continuously to ad. Prove that the component function u(x, y) must attain its minimum value on aD unless u(x,y) is a constant function. (Hint: Consider the modulus of analytic function g(z) = ef(x), and apply the result in problem 5)
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is...
6. Given a finite set A, denote IA] as a nurnber of elements in A. Let f : X → Y be a function with |XI, Yl< oo, i.e. X, Y are finite sets. Prove the following statements a) IXIS IYİ if f is injective. b) IY1S 1X1 if f is surjective. 6. Given a finite set A, denote IA] as a nurnber of elements in A. Let f : X → Y be a function with |XI, Yl
6. Let f:A B be a function with domain A and codomain B. Let S and T be subsets of the domain A a) Prove: f(ST)cf(S)n f(T) b) Give an example to show it is possible that f(SOT) f(S)nf (T). Name the domain, codomain, function, and sets S and T c) Let U and V be subsets of the codomain B. Prove: f (Unv)= f"(U)nfV)
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.