Q4Mathmetical reasoning. Please with full explanation and steps. I struggle with this topic very much. Explanation and typed answers will be appreciated. Thanks！ 1. Let f : X →Y be a function.(a) Prove that f(B1 N B2) = f(B1) n f(B2) for any B1, B2, E P(Y).(b) Prove that f(A N A2) s f(A1)n f (A2) for any A1, A2 E P(X). Give an exampleof a specific function f and sets A1, A, such that f(A1) n f(A2) Z f(AN Az2)2. Let f : X →Y be a function.(a) Prove that f is injective + f is injective + f is surjective.(b) Prove that f is surjective + f is surjective + f is injective.(Hint: when proving statements of the form P Q R, it is often easier to proveP = Q = R → P than to prove P Q and Q R)3. Prove Corollary 10.2.2 from class: for any n E Z4, if X1,. .. , Xn are finite sets whichare pairwise disjoint, thenUx =EX:|.i=1i=1n(Recall that U X; = X1 U X, U ...UX,{x]x € X; for some i e {1,2,..,n}} byi=1definition).4. Suppose X is a finite set and A, B C X. Show that if |A| + |B| > |X| + 3, then|AN B| > 3.5. Prove the following theorem from class:Theorem 1. Suppose that X and Y are nonempty finite sets with |X| < [Y| andf : X → Y is a function. Then f is not a surjection.6. Prove the following theorem from class:Theorem 2. Suppose that X and Y are finite sets with |X| = |Y\. Then a functionf: X → Y is injective if and only if it is surjective.

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Answered: 4. Suppose X is a finite set and A, B C…

Math

Advanced Math

4. Suppose X is a finite set and A, B C X. Show that if |A| + |B| > |X|+ 3, then |AN B| 2 3.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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