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7.S: Equivalence Relations (Summary)

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Important Definitions

• Relation from $$A$$ to $$B$$, page 364
• Relation on $$A$$, page 364
• Domain of a relation, page 364
• Range of a relation, page 364
• Inverse of a relation, page 373
• Reflexive relation, page 375
• Symmetric relation, page 375
• Transitiverelation,page375
• Equivalence relation, page 378
• Equivalence class, page 391
• Congruence class, page 392
• Partition of a set, page 395
• Integers modulo n, page 402
• Addition in $$\mathbb{Z}_n$$, page 404
• Multiplication in $$\mathbb{Z}_n$$, page 404

Important Theorems and Results about Relations, Equivalence Relations, and Equivalence Classes

• Theorem 7.6. Let $$R$$ be a relation from the set $$A$$ to the set $$B$$. Then

1. The domain of $$R^{-1}$$ is range of $$R$$. That is, dom($$R^{-1}$$) = range($$R$$).
2. The range of $$R^{-1}$$ is domain of $$R$$. That is, range($$R^{-1}$$) = dom($$R$$).
3. The inverse of $$R^{-1}$$ is $$R$$. That is, $$(R^{-1})^{-1} = R$$.
• Theorem 7.10. Let $$n \in \mathbb{N}$$ and let $$a, b \in \mathbb{Z}$$. Then $$a \equiv b$$ (mod $$n$$ if and only if $$a$$ and $$b$$ have the same remainder when divided by $$n$$.
• Theorem 7.14. Let $$A$$ be a nonempty set and let $$\sim$$ be an equivalence relation on $$A$$.

1. For each $$a \in A$$, $$a \in [a]$$.
2. For each $$a, b \in A$$, $$a \sim b$$ if and only if $$[a] = [b]$$.
3. For each $$a, b \in A$$, $$[a] = [b]$$ or $$[a] \cap [b] = \emptyset$$.
• Corollary 7.16. Let $$n \in \mathbb{N}$$. For each $$a \in \mathbb{Z}$$, let [$$a$$] represent the congruence class of $$a$$ modulo $$n$$.

1. For each $$a \in \mathbb{Z}$$, $$a \in [a]$$.
2. For each $$a, b \in \mathbb{Z}$$, $$a \equiv b$$ (mod $$n$$) if and only if $$[a] = [b]$$.
3. For each $$a, b \in \mathbb{Z}$$, $$[a] = [b]$$ or $$[a] \cap [b] = \emptyset$$.
• Corollary 7.17. Let $$n \in \mathbb{N}$$. For each $$a \in \mathbb{Z}$$, let [$$a$$] represent the congruence class of $$a$$ modulo $$n$$.

1. $$\mathbb{Z} =  \cup  \cup  \cup \cdot\cdot\cdot \cup [n - 1]$$
2. For $$j, k \in \{0, 1, 2, ..., n - 1\}$$, if $$j \ne k$$, then $$[j] \cap [k] = \emptyset$$.
• Theorem 7.18. Let $$\sim$$ be an equivalence relation on the nonempty set $$A$$. Then the collection $$\mathcal{C}$$ of all equivalence classes determined by $$\sim$$ is a partition of the set $$A$$.