Consider the binary relation R = {(x,y),(y,x),(z,z)} 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

GATE 2009 · Discrete Mathematics · Relations · easy

Answer: R is symmetric (every pair has its reverse in R) but NOT antisymmetric (the mutual pair (x,y)/(y,x) with x != y violates antisymmetry). Answer: A. R is symmetric but NOT antisymmetric.

1. Test symmetry: Check each pair in R: (x,y) in R -> is (y,x) in R? YES. (y,x) in R -> is (x,y) in R? YES. (z,z) in R -> is (z,z) in R? YES. All pairs in R have their reverse in R, so R IS symmetric.
2. Test antisymmetry: We have (x,y) in R AND (y,x) in R, but x != y. This violates antisymmetry. So R is NOT antisymmetric.
3. Identify correct option: R is symmetric (from step 1) but NOT antisymmetric (from step 2). This matches option A: 'R is symmetric but NOT antisymmetric.'