Solution to Show that the following relations are Partial order relations. (a) R on the set of … - Sikademy
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Archangel Macsika

Show that the following relations are Partial order relations. (a) R on the set of integers Z, defined by, R = {(a, b) | a ≤ b} (b) R on the set of integers Z, defined by, R = {(a, b) | a ≥ b} (c) R on the set of positive integers N, defined by, R = {(a, b) | a divides b} [Note: It is divisibility relation] (d) R on the Power set of a set S, defined by, R = {(A, B) | A is a subset of B} [Note: It is set inclu￾sion relation]

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Let us show that the following relations are partial order relations.


(a) R on the set of integers \Z , defined by, R = \{(a, b) | a ≤ b\}.


Since a\le a for any a\in\Z, we conclude that (a, a)\in R for any a\in\Z, and hence the relation R is reflexive. If (a,b)\in R and (b,a)\in R, then a\le b and b\le a, and hence a=b. It follows that R is an antisymmetric relation. If (a,b)\in R and (b,c)\in R, then a\le b and b\le c, and thus a\le c. Therefore, R is a transitive relation. We conclude that R is a partial order relation.


(b) R on the set of integers \Z , defined by, R = \{(a, b) | a ≥ b\}.


Since a\ge a for any a\in\Z, we conclude that (a, a)\in R for any a\in\Z, and hence the relation R is reflexive. If (a,b)\in R and (b,a)\in R, then a\ge b and b\ge a, and hence a=b. It follows that R is an antisymmetric relation. If (a,b)\in R and (b,c)\in R, then a\ge b and b\ge c, and thus a\ge c. Therefore, R is a transitive relation. We conclude that R is a partial order relation.


(c) R on the set of positive integers \N, defined by, R = \{(a, b) | a \text{ divides }b\}.


Since a| a for any a\in\N, we conclude that (a, a)\in R for any a\in\N, and hence the relation R is reflexive. If (a,b)\in R and (b,a)\in R, then a| b and b| a. Therefore, a\le b and b\le a, and hence a=b. It follows that R is an antisymmetric relation. If (a,b)\in R and (b,c)\in R, then a| b and b| c, and thus a| c. Therefore, R is a transitive relation. We conclude that R is a partial order relation.


(d) R on the Power set of a set S, defined by, R = \{(A, B) | A \text{ is a subset of } B\}.


Since A\subset A for any A\subset S we conclude that (A, A)\in R for any A\subset S, and hence the relation R is reflexive. If (A,B)\in R and (B,A)\in R, then A\subset B and B\subset A, and hence A=B. It follows that R is an antisymmetric relation. If (A,B)\in R and (B,C)\in R, then A\subset B and B\subset C, and thus A\subset C. Therefore, R is a transitive relation. We conclude that R is

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Question ID: mtid-5-stid-8-sqid-2868-qpid-1484