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 example of a specific function...

Q4 Mathmetical reasoning. Please with full explanation and steps. I struggle with this topic very much. Explanation and typed answers will be appreciated. Thanks！

Extracted text: 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 example of 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 prove P = 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 which are pairwise disjoint, then Ux =EX:|. i=1 i=1 n (Recall that U X; = X1 U X, U ...UX, {x]x € X; for some i e {1,2,..,n}} by i=1 definition). 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| and f : 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 function f: x → y is injective if and only if it is surjective. [y|="" and="" f="" :="" 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="" function="" f:="" x="" →="" y="" is="" injective="" if="" and="" only="" if="" it="" is="">