Let S = {1, 2} and define ρ and µ be relations on S defined by
$ρ = \{(1, 1),(1, 2),(2, 1)\}$
$µ = \{(1, 1),(2, 2)\}$
Is $ρ$ symmetric? To determine if a relation is symmetric, we need to check if whenever (a, b) is in the relation, (b, a) is also in the relation. In this case $ρ$ is symmetric right? Since
but for $µ$, this is not symmetric right?
Also I am confused about anti-symmetry. here is my working:
For the relation ρ = {(1, 1), (1, 2), (2, 1)}:
(1, 2) is in ρ, but (2, 1) is also in ρ. Since (1, 2) ≠ (2, 1), ρ is not antisymmetric.
For the relation µ = {(1, 1), (2, 2)}:
(1, 1) is in µ, and (2, 2) is also in µ. Thus, µ is antisymmetric. But I am still confused about this.