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| Properties of Relations |
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| Consider the set A = {a, b, c, …} then any subset of A x A is called a relation in A. Let R be any relation in A so that R is a subset of the product set A x A. We will discuss the properties a relation may possess. |
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| A relation R in a set A is said to be reflexive if every element of the set A is related to itself. This is if aRa where a
Î A, or (a, a) Î R for each a
Î A |
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| For example, |
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| (i) a triangle 'is congruent to' itself. |
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| (ii) 5 is a multiple of' 5. |
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A relation R, is said to be transitive if a 'is related to' b and b 'is related to' c then a 'is related to' c,  (a, b)  R and   if aRb, bRc then aRc. |
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| For example, |
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| (i) If DABC 'is congruent to'
DDEF and DDEF 'is congruent to'
DXYZ, then DABC 'is congruent to'
DXYZ |
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| (ii) If Ram 'is the brother of' Laxman and Laxman 'is the brother of' Bharat then Ram 'is the brother of' Bharat. |
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A relation R, on a set A, is symmetric if aRb then bRa  if a 'is related to' b, b 'is related to' a for a, b
Î R. Also if (a, b)
Î R then (b, a) Î R |
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| For example, |
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| (i) If DABC 'is congruent to'
DXYZ, then DXYZ 'is congruent to'
DABC. |
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| (ii) If Mary 'is the sister of' Lucy, then Lucy 'is the sister of' Mary. |
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| If a relation is (i) reflexive, (ii) transitive and, (iii) symmetric, then it is called an equivalence relation. |
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| For example, |
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| (i) is congruent to (ii) is parallel to (iii) is similar to, are some examples of equivalence relations. |
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Ordered Pairs and Cartesian Product
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