Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > Let, R = {(a, b): a,b ∈ Z and (a + b) is...
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Let, R = {(a, b): a,b ∈ Z and (a + b) is even}, then R is
- a)Reflexive relation on Z
- b)Equivalence relation on Z
- c)transitive relation on Z
- d)None of the above
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Let, R = {(a, b): a,b ∈ Z and (a + b) is even}, then R isa)Reflex...
Here, R = {(a, b): a,b ϵ Z and (a + b) is even}
- Since,a + a = 2a, ,which is even, so R is reflexive.
- If a + b is even then b + a will also be even. So, R is symmetric.
- Let, a = 3, b = 5, and c = 7.
a + b = 3 + 5 = 8 (which is even), b + c = 5 + 7 = 12 (which is again even) and a + c = 3 + 7 = 10 (which is also even)
Therefore, R is an equivalence relation on Z.
Hence, option (2) is correct.
Therefore, R is an equivalence relation on Z.
Hence, option (2) is correct.
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Let, R = {(a, b): a,b ∈ Z and (a + b) is even}, then R isa)Reflex...
Understanding the Relation R
R = {(a, b): a, b ∈ Z and (a + b) is even} defines a relation on the set of integers Z based on the evenness of the sum of two integers.
Evaluating Properties of R
1. Reflexive Relation:
- A relation is reflexive if for every element a in Z, (a, a) is in R.
- Here, (a + a) = 2a, which is always even.
- Thus, R is reflexive.
2. Symmetric Relation:
- A relation is symmetric if for every (a, b) in R, (b, a) is also in R.
- Since (a + b) is even, (b + a) = (a + b) is also even.
- Therefore, R is symmetric.
3. Transitive Relation:
- A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) is also in R.
- If (a + b) is even and (b + c) is even, then:
- a + b = 2m (even)
- b + c = 2n (even)
- Adding these gives (a + c) + 2b = 2m + 2n, which is even, indicating (a + c) is even.
- Thus, R is transitive.
Conclusion: Equivalence Relation
Since R is reflexive, symmetric, and transitive, it satisfies all properties of an equivalence relation. Thus, the correct answer is option 'B', indicating that R is an equivalence relation on Z.
This classification helps in understanding the structure of integers when grouped by evenness of sums, providing insight into their mathematical behavior.
R = {(a, b): a, b ∈ Z and (a + b) is even} defines a relation on the set of integers Z based on the evenness of the sum of two integers.
Evaluating Properties of R
1. Reflexive Relation:
- A relation is reflexive if for every element a in Z, (a, a) is in R.
- Here, (a + a) = 2a, which is always even.
- Thus, R is reflexive.
2. Symmetric Relation:
- A relation is symmetric if for every (a, b) in R, (b, a) is also in R.
- Since (a + b) is even, (b + a) = (a + b) is also even.
- Therefore, R is symmetric.
3. Transitive Relation:
- A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) is also in R.
- If (a + b) is even and (b + c) is even, then:
- a + b = 2m (even)
- b + c = 2n (even)
- Adding these gives (a + c) + 2b = 2m + 2n, which is even, indicating (a + c) is even.
- Thus, R is transitive.
Conclusion: Equivalence Relation
Since R is reflexive, symmetric, and transitive, it satisfies all properties of an equivalence relation. Thus, the correct answer is option 'B', indicating that R is an equivalence relation on Z.
This classification helps in understanding the structure of integers when grouped by evenness of sums, providing insight into their mathematical behavior.
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Let, R = {(a, b): a,b ∈ Z and (a + b) is even}, then R isa)Reflexive relation on Zb)Equivalence relation on Zc)transitive relation on Zd)None of the aboveCorrect answer is option 'B'. Can you explain this answer?
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