More detail question please see “picture” Total have 9 Question and need detail

More detail question please see “picture” Total have 9 Question and need detail step for solution.EXAMPLE——–Problem 1. Let f : X → Y be a function. For a subset A ⊆ X, we define
f(A) = {f(x) : x ∈ A}.
(1) Show that for A, B ⊆ X that f(A ∪ B) = f(A) ∪ f(B).
(2) Is it always true that f(A ∩ B) = f(A) ∩ f(B)? Prove this or find a counterexample. Problem 2. Let f : X → Y be a function. Show that f is a bijection exactly when
there is a function g : Y → X so that g(f(x)) = x for all x ∈ X and f(g(y)) = y for all
y ∈ Y .
Problem 3. For positive real numbers x and y, show that if xy > a, then either x > √
a
or y > √
a.
Problem 4. Show that at least one number among a1, . . . , an must be at least the
average value (a1 + · · · + an)/n. Problem 5. Show that √
3 is irrational.
Problem 6. Show that every natural number greater than 1 has a prime divisor without
using the fact that every such number admits a factorization into primes.
Requirements: need detail step for the solution(proof)

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