Practice Question 2: Union And Composition Of Equivalence Relations (1.1.4)

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6 sample questions

Question 1.
Define an equivalence relation.
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Think about each property individually.
Question 2.
What is the result of the union of two equivalence relations?
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Consider how pairs combine.
3Question 3.
Is the union of two equivalence relations always an equivalence relation?
- True
- False
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Think of specific pairs that could violate this.
4Question 4.
What property must hold for R1 ∪ R2 to be an equivalence relation?
- It must be reflexive
- It must be symmetric
- R1 ∘ R2 = R1 ∪ R2
- Both A and B
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Consider how both relations interact.
Question 5.
Prove the statement: If R1 is a subset of R2 and both are equivalence relations, what can you say about R1 ∪ R2 and R2?
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Consider subset relationships.
Question 6.
Construct two equivalence relations R1 and R2, where the union is not transitive. Provide elements that demonstrate this.
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Identify ordered pairs that fail to connect transitive links.

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