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Which of the following is not an equivalence relation on I, the set of integers ; x, y
- a)x R y x + y is an even integer
- b)x R y x = y
- c)x R y x < y
- d)x R y x – y is an even integer
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Which of the following is not an equivalence relation on I, the set of...
If R is a relation defined by xRy : ifx ⩽ y,then R is reflexive and transitive .But , it is not symmetric. Hence , R is not an equivalence relation.
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Which of the following is not an equivalence relation on I, the set of...
Equivalence Relation on Integers
An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive. Let's analyze each given relation to determine if it is an equivalence relation on the set of integers.
Option A: x R y if x + y is an even integer
- Reflexive: x + x = 2x, which is always even, so x R x is true.
- Symmetric: If x + y is even, then y + x is also even, so x R y implies y R x.
- Transitive: If x + y and y + z are even, then x + z is also even, so x R y and y R z implies x R z.
Therefore, this relation is an equivalence relation.
Option B: x R y if x = y
- Reflexive: x = x, so x R x is true.
- Symmetric: If x = y, then y = x, so x R y implies y R x.
- Transitive: If x = y and y = z, then x = z, so x R y and y R z implies x R z.
This relation is also an equivalence relation.
Option C: x R y if x < />
- Not Reflexive: x < x="" is="" not="" true="" for="" any="" integer="" x,="" so="" the="" relation="" is="" not="" />
Therefore, this relation is not an equivalence relation.
Option D: x R y if x - y is an even integer
- Reflexive: x - x = 0, which is even, so x R x is true.
- Symmetric: If x - y is even, then y - x is also even, so x R y implies y R x.
- Transitive: If x - y and y - z are even, then x - z is also even, so x R y and y R z implies x R z.
This relation is an equivalence relation.
Therefore, the correct answer is option C, as it does not satisfy the reflexivity property of an equivalence relation.
An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive. Let's analyze each given relation to determine if it is an equivalence relation on the set of integers.
Option A: x R y if x + y is an even integer
- Reflexive: x + x = 2x, which is always even, so x R x is true.
- Symmetric: If x + y is even, then y + x is also even, so x R y implies y R x.
- Transitive: If x + y and y + z are even, then x + z is also even, so x R y and y R z implies x R z.
Therefore, this relation is an equivalence relation.
Option B: x R y if x = y
- Reflexive: x = x, so x R x is true.
- Symmetric: If x = y, then y = x, so x R y implies y R x.
- Transitive: If x = y and y = z, then x = z, so x R y and y R z implies x R z.
This relation is also an equivalence relation.
Option C: x R y if x < />
- Not Reflexive: x < x="" is="" not="" true="" for="" any="" integer="" x,="" so="" the="" relation="" is="" not="" />
Therefore, this relation is not an equivalence relation.
Option D: x R y if x - y is an even integer
- Reflexive: x - x = 0, which is even, so x R x is true.
- Symmetric: If x - y is even, then y - x is also even, so x R y implies y R x.
- Transitive: If x - y and y - z are even, then x - z is also even, so x R y and y R z implies x R z.
This relation is an equivalence relation.
Therefore, the correct answer is option C, as it does not satisfy the reflexivity property of an equivalence relation.
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Which of the following is not an equivalence relation on I, the set of...
Option C is correct...
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Which of the following is not an equivalence relation on I, the set of integers ; x, ya)x R y x + y is an even integerb)x R y x = yc)x R y x < yd)x R y x – y is an even integerCorrect answer is option 'C'. Can you explain this answer?
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Which of the following is not an equivalence relation on I, the set of integers ; x, ya)x R y x + y is an even integerb)x R y x = yc)x R y x < yd)x R y x – y is an even integerCorrect answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following is not an equivalence relation on I, the set of integers ; x, ya)x R y x + y is an even integerb)x R y x = yc)x R y x < yd)x R y x – y is an even integerCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is not an equivalence relation on I, the set of integers ; x, ya)x R y x + y is an even integerb)x R y x = yc)x R y x < yd)x R y x – y is an even integerCorrect answer is option 'C'. Can you explain this answer?.
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