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Consider R and S be two equivalence relations, which of the following is true regarding the R and S
- a)The Largest equivalence relation which is inside R and S is R∩S
- b)The smallest equivalence relation which is inside R and S is R∩S
- c)R∩S can never be an equivalence relation
- d)R ∪ S is always an equivalence relation
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Consider R and S be two equivalence relations, which of the following ...
Option D is wrong as transitivity may break on taking the union of two equivalence relations
For two equivalence relation R and S
For two equivalence relation R and S
- The largest equivalence relation in R and S is R∩S
- The smallest equivalence relation which contains R and S is (R∪S)∞ {transitive closure}
So option A is correct.
Option C is trivially false as option A is correct.
Option C is trivially false as option A is correct.
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Consider R and S be two equivalence relations, which of the following ...
Understanding Equivalence Relations
Equivalence relations are binary relations that are reflexive, symmetric, and transitive. When dealing with two equivalence relations R and S on a set, we can analyze their intersections and unions for specific properties.
Intersection of R and S
- The intersection R ∩ S consists of all pairs (a, b) that are in both R and S.
- Since R and S are equivalence relations, they satisfy reflexivity, symmetry, and transitivity individually.
Why is R ∩ S the Largest Equivalence Relation Inside R and S?
- Reflexivity: For all elements a in the set, (a, a) is in both R and S, thus (a, a) is in R ∩ S.
- Symmetry: If (a, b) is in R ∩ S, then (a, b) is in both R and S. Hence, (b, a) is also in both R and S, thus (b, a) is in R ∩ S.
- Transitivity: If (a, b) and (b, c) are in R ∩ S, then they are in both R and S. Since R and S are transitive, we have (a, c) in both R and S, which means (a, c) is in R ∩ S.
Conclusion
Since R ∩ S satisfies reflexivity, symmetry, and transitivity, it is indeed an equivalence relation. Moreover, it contains only those pairs that are common to both R and S, making it the largest equivalence relation contained within both.
Verification of Other Options
- Option B: R ∩ S is not the smallest equivalence relation since it may not include all elements that can be equated under other relations.
- Option C: R ∩ S can indeed be an equivalence relation, as demonstrated above.
- Option D: R ∪ S is not guaranteed to be an equivalence relation because it may violate transitivity.
Thus, the correct statement is that the largest equivalence relation that is inside R and S is indeed R ∩ S.
Equivalence relations are binary relations that are reflexive, symmetric, and transitive. When dealing with two equivalence relations R and S on a set, we can analyze their intersections and unions for specific properties.
Intersection of R and S
- The intersection R ∩ S consists of all pairs (a, b) that are in both R and S.
- Since R and S are equivalence relations, they satisfy reflexivity, symmetry, and transitivity individually.
Why is R ∩ S the Largest Equivalence Relation Inside R and S?
- Reflexivity: For all elements a in the set, (a, a) is in both R and S, thus (a, a) is in R ∩ S.
- Symmetry: If (a, b) is in R ∩ S, then (a, b) is in both R and S. Hence, (b, a) is also in both R and S, thus (b, a) is in R ∩ S.
- Transitivity: If (a, b) and (b, c) are in R ∩ S, then they are in both R and S. Since R and S are transitive, we have (a, c) in both R and S, which means (a, c) is in R ∩ S.
Conclusion
Since R ∩ S satisfies reflexivity, symmetry, and transitivity, it is indeed an equivalence relation. Moreover, it contains only those pairs that are common to both R and S, making it the largest equivalence relation contained within both.
Verification of Other Options
- Option B: R ∩ S is not the smallest equivalence relation since it may not include all elements that can be equated under other relations.
- Option C: R ∩ S can indeed be an equivalence relation, as demonstrated above.
- Option D: R ∪ S is not guaranteed to be an equivalence relation because it may violate transitivity.
Thus, the correct statement is that the largest equivalence relation that is inside R and S is indeed R ∩ S.
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Consider R and S be two equivalence relations, which of the following is true regarding the R and Sa)The Largest equivalence relation which is inside R and S is R∩Sb)The smallest equivalence relation which is inside R and S is R∩Sc)R∩S can never be an equivalence relationd)R ∪ S is always an equivalence relationCorrect answer is option 'A'. Can you explain this answer?
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