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Consider the binary relation: R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE?
- a)R is symmetric but NOT antisymmetric
- b)R is NOT symmetric but antisymmetric
- c)R is both symmetric and antisymmetric
- d)R is neither symmetric nor antisymmetric
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Consider the binary relation: R = {(x, y), (x, z), (z, x), (z, y)} on ...
∴ R is not antisymmetric.
R is neither-symmetric nor anti-symmetric.
Most Upvoted Answer
Consider the binary relation: R = {(x, y), (x, z), (z, x), (z, y)} on ...
Question: Consider the binary relation: R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE?
Correct answer: Option D) R is neither symmetric nor antisymmetric
Explanation:
To determine whether the relation R is symmetric or antisymmetric, let's define these concepts first:
Symmetric Relation: A relation R on a set A is said to be symmetric if for every (a, b) ∈ R, (b, a) ∈ R.
Antisymmetric Relation: A relation R on a set A is said to be antisymmetric if for every (a, b) ∈ R and (b, a) ∈ R, a = b.
Now, let's analyze the given relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}.
Symmetric:
To check if R is symmetric, we need to verify if for every (a, b) ∈ R, (b, a) ∈ R.
In this case, we have (x, y) ∈ R, but (y, x) ∉ R. Similarly, (z, x) ∈ R, but (x, z) ∉ R. Therefore, R is NOT symmetric.
Antisymmetric:
To check if R is antisymmetric, we need to verify if for every (a, b) ∈ R and (b, a) ∈ R, a = b.
In this case, we have (x, z) ∈ R and (z, x) ∈ R, but x ≠ z. Therefore, R is NOT antisymmetric.
Since R fails to satisfy both the conditions for symmetry and antisymmetry, the correct answer is option D) R is neither symmetric nor antisymmetric.
Correct answer: Option D) R is neither symmetric nor antisymmetric
Explanation:
To determine whether the relation R is symmetric or antisymmetric, let's define these concepts first:
Symmetric Relation: A relation R on a set A is said to be symmetric if for every (a, b) ∈ R, (b, a) ∈ R.
Antisymmetric Relation: A relation R on a set A is said to be antisymmetric if for every (a, b) ∈ R and (b, a) ∈ R, a = b.
Now, let's analyze the given relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}.
Symmetric:
To check if R is symmetric, we need to verify if for every (a, b) ∈ R, (b, a) ∈ R.
In this case, we have (x, y) ∈ R, but (y, x) ∉ R. Similarly, (z, x) ∈ R, but (x, z) ∉ R. Therefore, R is NOT symmetric.
Antisymmetric:
To check if R is antisymmetric, we need to verify if for every (a, b) ∈ R and (b, a) ∈ R, a = b.
In this case, we have (x, z) ∈ R and (z, x) ∈ R, but x ≠ z. Therefore, R is NOT antisymmetric.
Since R fails to satisfy both the conditions for symmetry and antisymmetry, the correct answer is option D) R is neither symmetric nor antisymmetric.
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Consider the binary relation: R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE?a)R is symmetric but NOT antisymmetricb)R is NOT symmetric but antisymmetricc)R is both symmetric and antisymmetricd)R is neither symmetric nor antisymmetricCorrect answer is option 'D'. Can you explain this answer?
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