Consider the binary relation R = {(x, y), (x, z), (z, x), (z, y)} on the set…
2009
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
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Correct answer: D
Answer: The relation R is neither symmetric nor antisymmetric.
Symmetry check: For R to be symmetric, every time (a,b) is in R the pair (b,a) must also be in R. Here (x,y) is in R but (y,x) is not, and (z,y) is in R but (y,z) is not. Therefore R is not symmetric.
Antisymmetry check: A relation is antisymmetric if whenever (a,b) and (b,a) are both in R then a = b. Here (x,z) and (z,x) are both in R while x ≠ z, which violates antisymmetry. Thus R is not antisymmetric.
Conclusion: R fails both properties, so it is neither symmetric nor antisymmetric.
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