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| Types of Relations |
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| Let R be a relation on a set A. Then R is said to be |
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| Equivalence Relation |
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| A relation which is reflexive, symmetric and transitive is called the Equivalence relation. |
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| i.e., A relation R in a set A is called equivalence if it satisfies the following conditions. |
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| Remarks: |
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 The smallest equivalence relation in set A is the relation of equality in A. |
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 The largest equivalence relation in A is A x A. |
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| Inverse Relation |
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Let R Í A x B
be a relation from A to B. Then the inverse of R, denoted by R-1
is a relation from B to A, defined as  |
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| Clearly, Domain of R-1=
Range of R and Range of R-1= Domain of R. |
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| Example 1: |
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| If R = {(1,2), (1,5),(2,4),(3,5)} |
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| Domain of R-1=
{2, 5, 4} = Range of R |
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| Range of R-1=
{1, 2, 3} = Domain of R |
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| Example 2: |
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| If f is a relation from a set A to set B
such that f Í f-1 then, prove that f =
f-1. |
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| Suggested answer: |
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| From (i) and (ii), |
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 On the set of positive integers Z+, the relation “a divides b” is |
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 reflexive |
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 not symmetric |
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 transitive |
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 On the set of positive integers Z+, the relation "a |
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 not reflexive |
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 not symmetric |
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 transitive |
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 On the set of triangles, the relation “similar to” is |
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 reflexive |
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 symmetric |
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 transitive |
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 On the set of lines in a plane, the relation “parallel to” is |
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 reflexive |
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 symmetric |
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 transitive |
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 On the set of lines in a plane, the relation “perpendicular to” is |
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 not reflexive |
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 symmetric |
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 not transitive |
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 On the universal set of sets, the relation “subset of” is |
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 reflexive |
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 not symmetric |
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 transitive |
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 On the universal set of sets, the relation “subset of” is an |
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| antisymmetric relation. |
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 On the set of integers, the relation “congruence” is an equivalence relation. |
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