Let f : A → B be a function between sets A and B. Let X1, X2 ⊆ A and Y1, Y2 ⊆ B. a) Prove that if X1 ⊆ X2, then f(X1) ⊆ f(X2). b) Prove that if Y1 ⊆ Y2, then f -1 (Y1) ⊆ f -1(Y2)

Let A be a nonempty set and let f : A –→ B be a function. Prove that f is one-to-one if and only if there exists a function g : B → A so that gof = 14.

Let f: X → Y. Prove that f is one to one if B = f -1(f(B)) for all B ⊆ X.

Let f : R → R be such that for all x, y ∈ R, |f(x) - f(y)| ≤ (x - y)2. Prove that f is a constant function.

1. Let A = {1,2,3}, B = B = {4,5,6}, The functions f : A → C and g: B → C are defined by f(1) = 7, f(2)= 9, f(3) = 7, Does there exist a function h: A → B such that go h = f? Does there exist a function k : B → A such that fo k = g? Justify your answers. {4,5,6}, C = {7, 8, 9, 10}. g(4) = 10, g(5) = 7, g(6) = 9.

Let ℓ be a line and let f : ℓ → R be a function such that PQ = |f(P) - f(Q)| for every P, Q ∈ ℓ. Prove that f is a coordinate function for ℓ.

Let V be the set of functions f: R → R. For any two functions f, g in V, define the sum f + g to be the function given by (f + g)(x) = f(x) + g(x) for all real numbers. For any real number c and any function f in V, define scalar multiplication cf by (cf)(x) = cf(x) for all real numbers . Answer the following questions as partial verification that V is a vector space. (Addition is commutative:) Let f and g be any vectors in V. Then f(x) + g(x). since adding the real numbers f(x) and g(x) is a commutative operation. (A zero vector exists:) The zero vector in V is the function f given by f(x) = for all. (Additive inverses exist:) The additive inverse of the function f in V is a function g that satisfies f(x) + g(x) = 0 for all real numbers . The additive inverse of f is the function g() for all . me for all real numbers a (Scalar multiplication distributes over vector addition:) If c is any real number and fand g are two vectors in V, then c(f+g)(x) = c(f(x) + g(x)) =