Project ForgeDiscrete Mathematics
Set Theory & Abstract Algebra
GATE 2009
Medium
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?
- AR is symmetric but NOT antisymmetric.
- BR is NOT symmetric but antisymmetric.
- CR is both symmetric and antisymmetric.
- R is neither symmetric nor antisymmetric.
Explanation
Symmetric requires: for every (a,b) in R, (b,a) is too. (x,y) is in R but (y,x) is not — so R is NOT symmetric. Antisymmetric requires: whenever both (a,b) and (b,a) are in R, a=b. But (x,z) and (z,x) are both in R with x≠z — so R is NOT antisymmetric either. Hence neither.
Source: GATE CS 2009 (2009)
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