Assignment 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  

Which of the following relations RI, R2 and R3 is/are asymmetric?

(i). (x, y) e RI means x is married to y.

(ii)    (x, y) E R2 means x is the parent of y.

(iii)    (x, y) e R3 means x divides y, where x and y are positive integers.

Answer 1

i) R1 ("is married to") is symmetric and not asymmetric, because if x is married to y, then y is married to x, So it is not asymmetric. (There is no pair (x, y) for which (x, y) is in R1 but (y, x) is not in R1.

ii) R2 ("is the parent of") is asymmetric, since for every x and y for which x is the parent of y, y is not the parent of x. That is, for (x, y) is in R2, (y, x) is not in R2.

iii) R3 ("divides") is not asymmetric. Chose y = x, Then x divides y (which is the same as x) but also y (which is the same as x) divides x. That is, for these x and y, (x, y) is in R3, and (y, x) is also in R3.