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Homework 1

Math 447: Real Variables

Exercise 1 Let X be any set. Consider the two conditions below:

(a) Every injective function X → X is surjective.

(b) For every proper subset Y of X there is no surjection Y → X.

(Note: a proper subset Y of X is a set such that y ∈ Y =⇒ y ∈ X and ∃x ∈ X such that x /∈ Y .)

Prove that the two conditions above are equivalent. That is, prove (a) ⇐⇒ (b).

A set S is said to be “finite” if either of these equivalent conditions hold.

Hint: For (a) =⇒ (b) try the contrapositive. You may assume the “axiom of choice”.

Exercise 2 As in the lectures, let N be a set such that there is a function S : N → N with the following properties:

N1) ∃1 ∈ N.

N2) Im(S) = N \ {1}.

N3) If A ⊂ N is a set such that

S(A) ⊂ A and 1 ∈ A,

then A = N.

(a) Prove that if S is injective then N is not finite.

(b) Prove that property N3 implies the principle of mathematical induction:

Let P(n) be a proposition whose truth value depends on the value n ∈ N. Suppose P(1) is true and that P(n) =⇒ P (S(n)) for all n ∈ N. Then P(n) is true for al n ∈ N.

Hint: Let A = { n ∈ N | P(n) is true }.