State whether each of the following relations is a partial o

State whether each of the following relations is a partial ordering, and explain
why or why not.
(a) “isFatherOf” on the set of people.
(b) “isAncestorOf” on the set of people.
(c) “isOlderThan” on the set of people.
(d) “isSisterOf” on the set of people.
(e) fha; bi; ha; ai; hb; aig on the set fa; bg.
(f) fh2; 1i; h1; 3i; h2; 3ig on the set f1; 2; 3g.

Solution

A realtion is called partial order if it is antisymmetric , transitive and either reflexive or irreflexive

(a) “isFatherOf” on the set of people - is not a partial order since it fails transitivity clause.

a isFatherOf b , b isFatherOf c ... so if transitivity has to hold true a shoulbBeFatherOf c which is not the case a isGrandFatherOf c

(b) “isAncestorOf” on the set of people - is a partial order realtion

transitive : a isAncestorOf b , b isAncestorOf c ... so if transitivity has to hold true a isAncestorOf c which is also true

anitsymmetric : a isAncestorOf b and b isAncestorOf a is only true if a=b .

(c) “isOlderThan” on the set of people.- is a partial order realtion

transitive : true.

anitsymmetric : true only if a=b.

(d) “isSisterOf” on the set of people.- is a partial order realtion

transitive : true. : a isSisterOf b , b isSisterOf c so a isSisterOf c - true

antisymmetric : true : as not for all a,b does aRb and bRa holds true. therefore is antisymmetric