Which of the following is an equevalence relation on the set (a, b, c} 7 O (a, a), (a, b), (b, a). (b, 6)} O (a, a), (a, b), (b, a), (b, b), (c, c)) O (a, a), (a, b). (b. 6), (b, c). (c, a), (c, c)} O (a,b), (a,c). (b, a), (b,c). (c, a). (c, b))
Which of the following is an equevalence relation on the set (a, b, c} 7 O (a, a), (a, b), (b, a). (b, 6)} O (a, a), (a, b), (b, a), (b, b), (c, c)) O (a, a), (a, b). (b. 6), (b, c). (c, a), (c, c)} O (a,b), (a,c). (b, a), (b,c). (c, a). (c, b))
Algebra and Trigonometry (6th Edition)
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ISBN:9780134463216
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Equivalence Relations on a Set**
Please choose which of the following relations is an equivalence relation on the set \(\{a, b, c\}\):
1. \(\{(a, a), (a, b), (b, a), (b, b)\}\)
2. \(\{(a, a), (a, b), (b, a), (b, b), (c, c)\}\)
3. \(\{(a, a), (a, b), (b, b), (b, a), (c, a), (c, c)\}\)
4. \(\{(a, b), (a, c), (b, a), (b, c), (c, a), (c, b)\}\)
### Explanation
In mathematics, an equivalence relation on a set must satisfy three properties:
1. **Reflexivity:** Every element is related to itself.
2. **Symmetry:** If an element is related to another, then the second element is related to the first.
3. **Transitivity:** If an element is related to a second element and the second element is related to a third, then the first element must be related to the third.
Please review the given options to determine which set of ordered pairs satisfies all three of these properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc899dcba-268c-4add-8443-07635494adc0%2F9dacae5c-3744-42b3-9af9-3c6389aa0edb%2Fcow4af7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Equivalence Relations on a Set**
Please choose which of the following relations is an equivalence relation on the set \(\{a, b, c\}\):
1. \(\{(a, a), (a, b), (b, a), (b, b)\}\)
2. \(\{(a, a), (a, b), (b, a), (b, b), (c, c)\}\)
3. \(\{(a, a), (a, b), (b, b), (b, a), (c, a), (c, c)\}\)
4. \(\{(a, b), (a, c), (b, a), (b, c), (c, a), (c, b)\}\)
### Explanation
In mathematics, an equivalence relation on a set must satisfy three properties:
1. **Reflexivity:** Every element is related to itself.
2. **Symmetry:** If an element is related to another, then the second element is related to the first.
3. **Transitivity:** If an element is related to a second element and the second element is related to a third, then the first element must be related to the third.
Please review the given options to determine which set of ordered pairs satisfies all three of these properties.
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