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If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z is
- a)reflexive and transitive but not symmetric
- b)symmetric and transitive but not reflexive
- c)reflexive and symmetric but not transitive
- d)an equivalence relation
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab >...
so it is reflexive
so, it is symmetric 
So, it is not transitive.
Most Upvoted Answer
If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab >...
Understanding the Relation R
The relation R on the set of integers Z is defined as R = {(a, b) : 1 + ab > 0}. We will analyze the properties of this relation: reflexivity, symmetry, and transitivity.
1. Reflexivity
- A relation is reflexive if for every a in Z, (a, a) ∈ R.
- Here, we check: 1 + a*a > 0, which simplifies to 1 + a^2 > 0.
- Since a^2 is always non-negative for all integers a, 1 + a^2 is always greater than 0.
- Therefore, R is reflexive.
2. Symmetry
- A relation is symmetric if whenever (a, b) ∈ R, then (b, a) ∈ R.
- We check: If 1 + ab > 0, does it imply 1 + ba > 0?
- Since ab = ba, this condition holds true.
- Thus, R is symmetric.
3. Transitivity
- A relation is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R must also hold.
- Consider (a, b) ∈ R implies 1 + ab > 0, and (b, c) ∈ R implies 1 + bc > 0.
- For transitivity to hold, we need to show 1 + ac > 0, but this is not guaranteed.
- For example, let a = 1, b = -1, and c = 1. Then 1 + ab = 0 and 1 + bc = 0, but 1 + ac = 2 > 0. The relation fails for some combinations.
- Therefore, R is not transitive.
Conclusion
- R is reflexive and symmetric but not transitive.
- Hence, the correct answer is option 'C': R is reflexive and symmetric but not transitive.
The relation R on the set of integers Z is defined as R = {(a, b) : 1 + ab > 0}. We will analyze the properties of this relation: reflexivity, symmetry, and transitivity.
1. Reflexivity
- A relation is reflexive if for every a in Z, (a, a) ∈ R.
- Here, we check: 1 + a*a > 0, which simplifies to 1 + a^2 > 0.
- Since a^2 is always non-negative for all integers a, 1 + a^2 is always greater than 0.
- Therefore, R is reflexive.
2. Symmetry
- A relation is symmetric if whenever (a, b) ∈ R, then (b, a) ∈ R.
- We check: If 1 + ab > 0, does it imply 1 + ba > 0?
- Since ab = ba, this condition holds true.
- Thus, R is symmetric.
3. Transitivity
- A relation is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R must also hold.
- Consider (a, b) ∈ R implies 1 + ab > 0, and (b, c) ∈ R implies 1 + bc > 0.
- For transitivity to hold, we need to show 1 + ac > 0, but this is not guaranteed.
- For example, let a = 1, b = -1, and c = 1. Then 1 + ab = 0 and 1 + bc = 0, but 1 + ac = 2 > 0. The relation fails for some combinations.
- Therefore, R is not transitive.
Conclusion
- R is reflexive and symmetric but not transitive.
- Hence, the correct answer is option 'C': R is reflexive and symmetric but not transitive.
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If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z isa)reflexive and transitive but not symmetricb)symmetric and transitive but not reflexivec)reflexive and symmetric but not transitived)an equivalence relationCorrect answer is option 'C'. Can you explain this answer?
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If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z isa)reflexive and transitive but not symmetricb)symmetric and transitive but not reflexivec)reflexive and symmetric but not transitived)an equivalence relationCorrect 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 If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z isa)reflexive and transitive but not symmetricb)symmetric and transitive but not reflexivec)reflexive and symmetric but not transitived)an equivalence relationCorrect 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 If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z isa)reflexive and transitive but not symmetricb)symmetric and transitive but not reflexivec)reflexive and symmetric but not transitived)an equivalence relationCorrect answer is option 'C'. Can you explain this answer?.
If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z isa)reflexive and transitive but not symmetricb)symmetric and transitive but not reflexivec)reflexive and symmetric but not transitived)an equivalence relationCorrect 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 If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z isa)reflexive and transitive but not symmetricb)symmetric and transitive but not reflexivec)reflexive and symmetric but not transitived)an equivalence relationCorrect 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 If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z isa)reflexive and transitive but not symmetricb)symmetric and transitive but not reflexivec)reflexive and symmetric but not transitived)an equivalence relationCorrect answer is option 'C'. Can you explain this answer?.
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