# Abstract Math

The detailed problems are attached. There are two parts of the problem. Please write the processes clearly.

1. Let A and B be sets.

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(a) Prove that if A is infi nite and |A| = |B|, then B is infinite.

(b) If A is countably innite and B is finite, use induction on |B| to prove that A U B is countably infinite.

2. Let A be a countably innite set. Prove that:

(a) The set {B ⊆A | |B| = 1} of one-element subsets of A is countably infinite.

2. Prove that “has the same cardinality” is an equivalence relation (i.e. show that it is reflexive, symmetric, and transitive).

3. Let A be a finite set. If A has the same cardinality as both Nk and Nm, use the Pigeonhole Principle to show that k = m.

4. Let A and B be sets.

(a) If A is infinite and A⊆B, prove that B is infinite.

(b) If A and B are finite sets with |A| = |B| and f : A → B is an injective function, prove that f is also surjective. (Hint: You may use the statement of part (c) of this problem in your proof.)

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