# Questions tagged [equivalence-relations]

For questions about relations that are reflexive, symmetric, and transitive. These are relations that model a sense of "equality" between elements of a set. Consider also using the (relation) tag.

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### Show that $P$ is a partition of the corresponding set $A$ and define the equivalence relation induced by partition $P$.

$A = \mathbb N$ and $P = \{ \{ m \in \mathbb N: m \ \ \text{is a multiple of} \ \ 3\}, \{ m \in \mathbb N: m \ \ \text{is not a multiple of 3}\} \}$. How can I define the equivalence relation ...
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### Union of equivalence relation, problem If R ∪ S is an equivalence relation on A, then R and S are equivalence relations on A.

Let A be a non-empty set and let R and S be relations defined on A. If R ∪ S is an equivalence relation on A, then R and S are equivalence relations on A. how can i prove it? i think it is not ...
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### Equivalence relation proof, problem $x∼y \iff x^2 − 2x + 1 = y^2 + 4y + 4$

Determine if the following relationships are equivalent. If they are, determine the equivalence classes, give a set of indices and the quotient set. Sketch out the graph of each relationship (whether ...
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### Is $G=\{(x,y)\in \mathbb{Z}\times\mathbb{Z}:x+y<4\}$ irreflexive?

Let $G=\{(x,y)\in \mathbb{Z}\times\mathbb{Z}:x+y<4\}$ Is $G$ irreflexive? I know $<$ is an irreflexive relation. But how to show it? How do I do this? The Reflexive and Irreflexive property ...
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### List one of three properties of an equivalence relations

I have the following question and answers: Let $A = \{1,2,3,4\}$. Write down $R \subseteq A x A$ which has one of the three properties of an equivalence relation (3 cases). The official answers for ...
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### Equivalence Relations: Understanding Compatibility

Synopsis My textbook, near the end of the section on equivalence relations, mentions the problem of "defining functions on a quotient set". Specifically, assume that $R$ is an equivalence relation on ...