Find the inverse functions of the following functions if

Question

Part A-

1. Suppose f : A → B is a bijection. Prove that f^-1 is a bijection.

2. Let f : A → B and g : B → C. Prove or disprove:

(a) If g o f is a surjection, then f is a surjection.

(b) If f is a surjection, then g o f is a surjection.

(c) If g o f is an injection, then f is an injection.

(d) If f is an injection, then g o f is an injection.

3. Let f : A → B be a bijection. Prove that for all y ∈ B, (f o f^-1)(y) = y.

4. Find the inverse functions of the following functions. If the function is not invertible, explain why.

(a) f : R →  R defined by f(x) = (x³+1)/3.

(b) g : Z → R defined by g(x) = (x³+1)/3.

(c) h: R x R → R x R defined by h(x, y) = (2x + y, x + y).

Part B-

1. Let R⁺ = {x ∈ R|x > 0}. Prove that |R| = |R⁺|.

2. Let N^N = {f | f : N → N}.

a) Name two elements of N^N.

b) Prove that N₀ < |N^N|.

3. Let A and B be sets such that |A| = |B| = N₀. Prove or disprove:

a) |A ∩ B| = N₀.

b) |A ∪ B| = N₀.

c) The irrational numbers have cardinality N₀.

d) Let x be any number. |A x {x}| = N₀.