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Let R be a relation on the set of ordered pairs of positive integers such that ((p, q), (r, s)) ∈ R if and only if p–s = q–r. Which one of the following is true about R?
- a)Both reflexive and symmetric
- b)Reflexive but not symmetric
- c)Not reflexive but symmetric
- d)Neither reflexive nor symmetric
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let R be a relation on the set of ordered pairs of positive integers s...
((p, q), (r, s)) ∈ R if and only if p–s = q–r (p, q) is not related to (p, q) as p-q is not same as q-p.
The relation is symmetric because if p–s = q–r, then s-q = s-p.
The relation is symmetric because if p–s = q–r, then s-q = s-p.
Most Upvoted Answer
Let R be a relation on the set of ordered pairs of positive integers s...
Answer:
To determine whether the relation R is reflexive or symmetric, we need to understand the definitions of these properties in relation to the given relation.
Reflexive Property:
A relation R on a set A is said to be reflexive if every element of A is related to itself. In other words, for every element a in A, (a, a) must be in R.
Symmetric Property:
A relation R on a set A is said to be symmetric if for every pair of elements (a, b) in R, the pair (b, a) is also in R.
Analysis:
Let's analyze the relation R defined as ((p, q), (r, s)) R if and only if ps = qr.
Reflexive Property:
To determine if R is reflexive, we need to check if for every element (a, a) in R. In this case, we need to check if (p, p) R for all positive integers p.
If we substitute p = q = r = s = 1, we get ps = qr = 1. Therefore, (1, 1) R.
However, if we substitute p = q = r = s = 2, we get ps = qr = 4. Therefore, (2, 2) is not in R.
Since there exists at least one positive integer for which (a, a) is not in R, the relation R is not reflexive.
Symmetric Property:
To determine if R is symmetric, we need to check if for every pair of elements ((p, q), (r, s)) in R, the pair ((r, s), (p, q)) is also in R.
Let ((p, q), (r, s)) be in R. This means that ps = qr. To satisfy the symmetric property, we need to show that rs = pq.
By substituting p = r, q = s in the equation ps = qr, we get rs = pq. Therefore, ((r, s), (p, q)) is in R.
Since the relation R satisfies the symmetric property, the relation R is symmetric.
Conclusion:
Based on the analysis, we can conclude that the relation R is not reflexive but it is symmetric. Therefore, the correct answer is option 'C': Not reflexive but symmetric.
To determine whether the relation R is reflexive or symmetric, we need to understand the definitions of these properties in relation to the given relation.
Reflexive Property:
A relation R on a set A is said to be reflexive if every element of A is related to itself. In other words, for every element a in A, (a, a) must be in R.
Symmetric Property:
A relation R on a set A is said to be symmetric if for every pair of elements (a, b) in R, the pair (b, a) is also in R.
Analysis:
Let's analyze the relation R defined as ((p, q), (r, s)) R if and only if ps = qr.
Reflexive Property:
To determine if R is reflexive, we need to check if for every element (a, a) in R. In this case, we need to check if (p, p) R for all positive integers p.
If we substitute p = q = r = s = 1, we get ps = qr = 1. Therefore, (1, 1) R.
However, if we substitute p = q = r = s = 2, we get ps = qr = 4. Therefore, (2, 2) is not in R.
Since there exists at least one positive integer for which (a, a) is not in R, the relation R is not reflexive.
Symmetric Property:
To determine if R is symmetric, we need to check if for every pair of elements ((p, q), (r, s)) in R, the pair ((r, s), (p, q)) is also in R.
Let ((p, q), (r, s)) be in R. This means that ps = qr. To satisfy the symmetric property, we need to show that rs = pq.
By substituting p = r, q = s in the equation ps = qr, we get rs = pq. Therefore, ((r, s), (p, q)) is in R.
Since the relation R satisfies the symmetric property, the relation R is symmetric.
Conclusion:
Based on the analysis, we can conclude that the relation R is not reflexive but it is symmetric. Therefore, the correct answer is option 'C': Not reflexive but symmetric.
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Let R be a relation on the set of ordered pairs of positive integers such that ((p, q), (r, s)) ∈ R if and only if p–s = q–r. Which one of the following is true about R?a)Both reflexive and symmetricb)Reflexive but not symmetricc)Not reflexive but symmetricd)Neither reflexive nor symmetricCorrect answer is option 'C'. Can you explain this answer?
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