Assignement Question

Question

Let R be a relation on a set A.

R is antisymmetric iff Vx,y e A, if x R y and y R x, then x = y.	or, equivalently:	R is antisymmetric iff Vx,y E A, if (x, y) E R and (y, x) E R, then x = y.
R is asymmetric iff Vx,y E A, if x R y, then y R x.	or, equivalently:	R is asymmetric iff Vx,y E A, if (x, y) E R, then (y, x)	R.

Question:

Lets = {a, b, c). The following three relations are defined on S as follows:

•                  Tl = { (a,a), (a, b), (b, c), (c, b) }

•                  T2 = { (a, a), (b, b), (a, b), (c, b) }

•        T3={ (a,b), (b, c), (c, a) }

Indicate whether each of these three relations are antisymmetric or asymmetric by filling in the table below with a tick (V) or cross (x) in each cell.   

Relation	Antisymmetric	Asymmetric

Answer 1

Often times, it's easier to prove when these are properties are violated.

T1 is not antisymmetric, as (c, b) and (b, c) exist in it. For every other relation, there is no pair (x, y) and (y, x) where x != y. So T2 and T3 are antisymmetric.

I think you're missing a not symbol (¬) in that definition of asymmetric. Asymmetry is the same as antisymmetry except that there's no x != y condition, meaning that a pair (x, x) cannot exist in an asymmetric set. T1 is already eliminated since it is not antisymmetric, but T2 is also eliminated as well since it contains (a, a). T3 does not contain any pairs like that, so it is asymmetric as well as antisymmetric.

 

Relation	Antisymmetric	Asymmetric
T1	x	x
T2	v	x
T3	v	v