Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

1.

Let $a,b\in \mathbb{Z}$ and let $n\in \mathbb{N}\text{.}$ On Definition of Section 3.1, we defined what it means to say that $a$ is congruent to $b$ modulo $n\text{.}$ Write this definition and state two different conditions that are equivalent to the definition.

2.

Explain why congruence modulo $n$ is a relation on $\mathbb{Z}\text{.}$

3.

Carefully review Theorem 3.36 and the proofs given on Theorem 3.36 of Section 3.5. In terms of the properties of relations introduced in Beginning Activity 1, what does this theorem say about the relation of congruence modulo $n$ on the integers?

4.

Write a complete statement of Theorem 3.37 and Corollary 3.38.

5.

Write a proof of the symmetric property for congruence modulo $n\text{.}$ That is, prove the following:

Let $a,b\in \mathbb{Z}$ and let $n\in \mathbb{N}\text{.}$ If $a\equiv b(\mathrm{mod}n)\text{,}$ then $b\equiv a(\mathrm{mod}n)\text{.}$