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A relation on the integers 0 through 4 is defined by:
R = {(x, y): x + y ≤ 2x}
Which of the properties listed below applies to this relation?
I. Transitivity
II. Symmetry
III. Reflexivity
- a)II and III
- b)I and III
- c)III only
- d)I only
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A relation on the integers 0 through 4 is defined by:R = {(x, y): x +...
Relation R on integer {0, 1, 2, 3, 4} is defined as
R = {(x, y) x + y ≤ 2x}
Reflexive relations : Reflexive relation on the set is a binary element in which every element is related to itself.
Transitive relation: Let A be a set in which the relation R defined. R is said to be transitive, if
(a, b) ∈ R and (b, a) ∈ R ⇒ (a, c) ∈ R,
That is aRb and bRc ⇒ aRc where a, b, c ∈ A
symmetric relation: Let A be a set in which the relation R defined. Then R is said to be a symmetric relation, if (a, b) ∈ R ⇒ (b, a) ∈ R, that is, aRb ⇒ bRa for all (a, b) ∈ R.
So, possible set of elements are
R = { (0, 0), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2), (3, 3), (4, 0), (4, 1), (4, 3), (4, 4)}
From the set it is clear that relation is reflexive and transitive but it is not symmetric.
Most Upvoted Answer
A relation on the integers 0 through 4 is defined by:R = {(x, y): x +...
Properties of the Given Relation:
Reflexivity:
- A relation R on a set S is reflexive if for every element x in S, (x, x) belongs to R.
- In this case, for every integer x in the set {0, 1, 2, 3, 4}, we have x + x ≤ 2x, which is always true.
- Therefore, the relation R is reflexive.
Transitivity:
- A relation R on a set S is transitive if for all x, y, and z in S, if (x, y) belongs to R and (y, z) belongs to R, then (x, z) must also belong to R.
- Let's consider x, y, and z such that (x, y) and (y, z) belong to R.
- This means x + y ≤ 2x and y + z ≤ 2y.
- Adding these two inequalities, we get x + y + y + z ≤ 2x + 2y, which simplifies to x + z ≤ 2x.
- Hence, (x, z) belongs to R, and the relation R is transitive.
Conclusion:
- The relation R is reflexive and transitive, but it is not symmetric since x + y is not necessarily equal to y + x.
- Therefore, the correct option is B) I and III.
Reflexivity:
- A relation R on a set S is reflexive if for every element x in S, (x, x) belongs to R.
- In this case, for every integer x in the set {0, 1, 2, 3, 4}, we have x + x ≤ 2x, which is always true.
- Therefore, the relation R is reflexive.
Transitivity:
- A relation R on a set S is transitive if for all x, y, and z in S, if (x, y) belongs to R and (y, z) belongs to R, then (x, z) must also belong to R.
- Let's consider x, y, and z such that (x, y) and (y, z) belong to R.
- This means x + y ≤ 2x and y + z ≤ 2y.
- Adding these two inequalities, we get x + y + y + z ≤ 2x + 2y, which simplifies to x + z ≤ 2x.
- Hence, (x, z) belongs to R, and the relation R is transitive.
Conclusion:
- The relation R is reflexive and transitive, but it is not symmetric since x + y is not necessarily equal to y + x.
- Therefore, the correct option is B) I and III.
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A relation on the integers 0 through 4 is defined by:R = {(x, y): x + y ≤ 2x}Which of the properties listed below applies to this relation?I. TransitivityII. SymmetryIII. Reflexivitya)II and IIIb)I and IIIc)III onlyd)I onlyCorrect answer is option 'B'. Can you explain this answer?
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A relation on the integers 0 through 4 is defined by:R = {(x, y): x + y ≤ 2x}Which of the properties listed below applies to this relation?I. TransitivityII. SymmetryIII. Reflexivitya)II and IIIb)I and IIIc)III onlyd)I onlyCorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A relation on the integers 0 through 4 is defined by:R = {(x, y): x + y ≤ 2x}Which of the properties listed below applies to this relation?I. TransitivityII. SymmetryIII. Reflexivitya)II and IIIb)I and IIIc)III onlyd)I onlyCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A relation on the integers 0 through 4 is defined by:R = {(x, y): x + y ≤ 2x}Which of the properties listed below applies to this relation?I. TransitivityII. SymmetryIII. Reflexivitya)II and IIIb)I and IIIc)III onlyd)I onlyCorrect answer is option 'B'. Can you explain this answer?.
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