Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Consider the following relationsR1(a,b) iff (...
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Consider the following relations
R1(a,b) iff (a+b) is even over the set of integers
R2(a,b) iff (a+b) is odd over the set of integers
R3(a,b) iff a.b > 0 over the set of non-zero rational numbers
R4(a,b) iff |a - b| <= 2 over the set of natural numbers
R2(a,b) iff (a+b) is odd over the set of integers
R3(a,b) iff a.b > 0 over the set of non-zero rational numbers
R4(a,b) iff |a - b| <= 2 over the set of natural numbers
Q. Which of the following statements is correct?
- a)R1 and R2 are equivalence relations, R3 and R4 are not
- b)R1 and R3 are equivalence relations, R2 and R4 are not
- c)R1 and R4 are equivalence relations, R2 and R3 are not
- d)R1, R2, R3 and R4 are all equivalence relations
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider the following relationsR1(a,b) iff (a+b) is even over the set...
So basically, we have to tell whether these relations are equivalence or not.
- R1(a,b)
- Reflexive : Yes, because (a+a) is even.
- Symmetrix : Yes, (a+b) is even ⟹ (b+a) is even.
- Transitive : Yes, because (a+b) is even and (b+c) is even ⟹ (a+c) is even.
So R1 is equivalence relation.
2. R2(a,b)
- Reflexive : No, because (a+a) is even.
So R2 is not equivalence relation.
3. R3(a,b)
- Reflexive : Yes, because a.a > 0.
- Symmetrix : Yes, a.b > 0 ⟹ b.a > 0.
- Transitive : Yes, because a.b > 0 and b.c > 0 ⟹ a.c > 0.
So R3 is equivalence relation.
4. R4(a,b)
- Reflexive : Yes, because |a-a| ≤ 2.
- Symmetrix : Yes, |a-b| ≤ 2 ⟹ |b-a| ≤ 2.
- Transitive : No, because |a-b| ≤ 2 and |b-c| ≤ 2 ⇏ (a-c) is even.
So R4 is not equivalence relation.
Most Upvoted Answer
Consider the following relationsR1(a,b) iff (a+b) is even over the set...
R1 and R3 are equivalence relations, R2 and R4 are not
To determine whether each relation is an equivalence relation, we need to check three properties: reflexiveness, symmetry, and transitivity.
R1(a,b) iff (a + b) is even over the set of integers:
- Reflexivity: For any integer a, (a + a) = 2a, which is always even. Therefore, R1 is reflexive.
- Symmetry: If (a + b) is even, then (b + a) is also even since addition is commutative. Therefore, R1 is symmetric.
- Transitivity: If (a + b) is even and (b + c) is even, then (a + c) is also even since addition is associative. Therefore, R1 is transitive.
R2(a,b) iff (a + b) is odd over the set of integers:
- Reflexivity: For any integer a, (a + a) = 2a, which is always even. Therefore, R2 is not reflexive.
- Symmetry: If (a + b) is odd, then (b + a) is also odd since addition is commutative. Therefore, R2 is symmetric.
- Transitivity: If (a + b) is odd and (b + c) is odd, then (a + c) is even, which violates the transitive property. Therefore, R2 is not transitive.
R3(a,b) iff a * b ≠ 0 over the set of non-zero rational numbers:
- Reflexivity: For any non-zero rational number a, a * a ≠ 0. Therefore, R3 is reflexive.
- Symmetry: If a * b ≠ 0, then b * a ≠ 0 since multiplication is commutative. Therefore, R3 is symmetric.
- Transitivity: If a * b ≠ 0 and b * c ≠ 0, then a * c ≠ 0 since multiplication is associative. Therefore, R3 is transitive.
R4(a,b) iff |a - b| = 2 over the set of natural numbers:
- Reflexivity: For any natural number a, |a - a| = 0, which is not equal to 2. Therefore, R4 is not reflexive.
- Symmetry: If |a - b| = 2, then |b - a| = 2 since the absolute value of a difference is the same regardless of the order. Therefore, R4 is symmetric.
- Transitivity: If |a - b| = 2 and |b - c| = 2, then |a - c| = 2 since the absolute value of a difference is always positive. Therefore, R4 is transitive.
Based on the analysis above, we can conclude that R1 and R3 are equivalence relations because they satisfy all three properties (reflexivity, symmetry, and transitivity). On the other hand, R2 and R4 are not equivalence relations because they fail to satisfy at least one of the properties. Hence, the correct answer is option 'B' - R1 and R3 are equivalence relations, R2 and R4 are not.
To determine whether each relation is an equivalence relation, we need to check three properties: reflexiveness, symmetry, and transitivity.
R1(a,b) iff (a + b) is even over the set of integers:
- Reflexivity: For any integer a, (a + a) = 2a, which is always even. Therefore, R1 is reflexive.
- Symmetry: If (a + b) is even, then (b + a) is also even since addition is commutative. Therefore, R1 is symmetric.
- Transitivity: If (a + b) is even and (b + c) is even, then (a + c) is also even since addition is associative. Therefore, R1 is transitive.
R2(a,b) iff (a + b) is odd over the set of integers:
- Reflexivity: For any integer a, (a + a) = 2a, which is always even. Therefore, R2 is not reflexive.
- Symmetry: If (a + b) is odd, then (b + a) is also odd since addition is commutative. Therefore, R2 is symmetric.
- Transitivity: If (a + b) is odd and (b + c) is odd, then (a + c) is even, which violates the transitive property. Therefore, R2 is not transitive.
R3(a,b) iff a * b ≠ 0 over the set of non-zero rational numbers:
- Reflexivity: For any non-zero rational number a, a * a ≠ 0. Therefore, R3 is reflexive.
- Symmetry: If a * b ≠ 0, then b * a ≠ 0 since multiplication is commutative. Therefore, R3 is symmetric.
- Transitivity: If a * b ≠ 0 and b * c ≠ 0, then a * c ≠ 0 since multiplication is associative. Therefore, R3 is transitive.
R4(a,b) iff |a - b| = 2 over the set of natural numbers:
- Reflexivity: For any natural number a, |a - a| = 0, which is not equal to 2. Therefore, R4 is not reflexive.
- Symmetry: If |a - b| = 2, then |b - a| = 2 since the absolute value of a difference is the same regardless of the order. Therefore, R4 is symmetric.
- Transitivity: If |a - b| = 2 and |b - c| = 2, then |a - c| = 2 since the absolute value of a difference is always positive. Therefore, R4 is transitive.
Based on the analysis above, we can conclude that R1 and R3 are equivalence relations because they satisfy all three properties (reflexivity, symmetry, and transitivity). On the other hand, R2 and R4 are not equivalence relations because they fail to satisfy at least one of the properties. Hence, the correct answer is option 'B' - R1 and R3 are equivalence relations, R2 and R4 are not.
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Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer?
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Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer?.
Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer?.
Solutions for Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Computer Science Engineering (CSE).
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Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Consider the following relationsR1(a,b) iff (a+b) is even over the set of integersR2(a,b) iff (a+b) is odd over the set of integersR3(a,b) iff a.b > 0 over the set of non-zero rational numbersR4(a,b) iff |a - b| <= 2 over the set of natural numbersQ.Which of the following statements is correct?a)R1 and R2 are equivalence relations, R3 and R4 are notb)R1 and R3 are equivalence relations, R2 and R4 are notc)R1 and R4 are equivalence relations, R2 and R3 are notd)R1, R2, R3 and R4 are all equivalence relationsCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice Computer Science Engineering (CSE) tests.
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