P(A),X r Y
X
Ì Y or X = Y.To show that r is a partial order relation, you will need to show that r is reflexive, antisymmetric and transitive.
5. Let A = {1, 2, 3, 4} and define the relation r
on the power set of A, P(A), as follows: for X, YP(A),
X r Y X
Ì Y or X = Y.
To show that r is a partial order relation, you will need to
show that r is reflexive, antisymmetric and transitive.
Suppose that X is an element of P(A). Now X = X, so X r X, by the definition of r. Thus the relation r is reflexive.
Suppose that X and Y are distinct elements of P(A) and that X r Y. Since X r Y and X and Y are not equal, we know that X Ì Y, that is, X is a proper subset of Y. If X is a proper subset of Y, then Y cannot possibly be a proper subset of X and hence Y is not related to X by r. Thus the relation r is antisymmetric.
Suppose that X, Y and Z are elements of P(A) and that X r Y and Y r Z. Since X r Y we know that X is a proper subset of Y or X is equal to Y. Since Y r Z, we know that Y is proper subset of Z or Y is equal to Z. If X, Y and Z are all equal, then X = Z. If X, Y and Z are not all equal, then X is a proper subset of Z. Therefore X r Z. Thus the relation r is transitive.
Therefore, the relation r is a partial order relation.