Partial Order RelationsA relation R on a set A is called a partial order relation if it satisfies the following three properties:
Example1: Show whether the relation (x, y) ∈ R, if, x ≥ y defined on the set of +ve integers is a partial order relation. Solution: Consider the set A = {1, 2, 3, 4} containing four +ve integers. Find the relation for this set such as R = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (1, 1), (2, 2), (3, 3), (4, 4)}. Reflexive: The relation is reflexive as for every a ∈ A. (a, a) ∈ R, i.e. (1, 1), (2, 2), (3, 3), (4, 4) ∈ R. Antisymmetric: The relation is antisymmetric as whenever (a, b) and (b, a) ∈ R, we have a = b. Transitive: The relation is transitive as whenever (a, b) and (b, c) ∈ R, we have (a, c) ∈ R. Example: (4, 2) ∈ R and (2, 1) ∈ R, implies (4, 1) ∈ R. As the relation is reflexive, antisymmetric and transitive. Hence, it is a partial order relation. Example2: Show that the relation 'Divides' defined on N is a partial order relation. Solution: Reflexive: We have a divides a, ∀ a∈N. Therefore, relation 'Divides' is reflexive. Antisymmetric: Let a, b, c ∈N, such that a divides b. It implies b divides a iff a = b. So, the relation is antisymmetric. Transitive: Let a, b, c ∈N, such that a divides b and b divides c. Then a divides c. Hence the relation is transitive. Thus, the relation being reflexive, antisymmetric and transitive, the relation 'divides' is a partial order relation. Example3: (a) The relation ⊆ of a set of inclusion is a partial ordering or any collection of sets since set inclusion has three desired properties:
(b) The relation ≤ on the set R of real no that is Reflexive, Antisymmetric and transitive. (c) Relation ≤ is a Partial Order Relation. nAry RelationsBy an nary relation, we mean a set of ordered ntuples. For any set S, a subset of the product set S^{n} is called an nary relation on S. In particular, a subset of S^{3} is called a ternary relation on S. Partial Order Set (POSET):The set A together with a partial order relation R on the set A and is denoted by (A, R) is called a partial orders set or POSET. Total Order RelationConsider the relation R on the set A. If it is also called the case that for all, a, b ∈ A, we have either (a, b) ∈ R or (b, a) ∈ R or a = b, then the relation R is known total order relation on set A. Example: Show that the relation '<' (less than) defined on N, the set of +ve integers is neither an equivalence relation nor partially ordered relation but is a total order relation. Solution: Reflexive: Let a ∈ N, then a < a As, the relation '<' (less than) is not reflexive, it is neither an equivalence relation nor the partial order relation. But, as ∀ a, b ∈ N, we have either a < b or b < a or a = b. So, the relation is a total order relation. Equivalence ClassConsider, an equivalence relation R on a set A. The equivalence class of an element a ∈ A, is the set of elements of A to which element a is related. It is denoted by [a]. Example: Let R be an equivalence relations on the set A = {4, 5, 6, 7} defined by Determine its equivalence classes. Solution: The equivalence classes are as follows: Circular RelationConsider a binary relation R on a set A. Relation R is called circular if (a, b) ∈ R and (b, c) ∈ R implies (c, a) ∈ R. Example: Consider R is an equivalence relation. Show that R is reflexive and circular. Solution: Reflexive: As, the relation, R is an equivalence relation. So, reflexivity is the property of an equivalence relation. Hence, R is reflexive. Circular: Let (a, b) ∈ R and (b, c) ∈ R Thus, R is Circular. Compatible RelationA binary relation R on a set A that is Reflexive and symmetric is called Compatible Relation. Every Equivalence Relation is compatible, but every compatible relation need not be an equivalence. Example: Set of a friend is compatible but may not be an equivalence relation. Friend Friend
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