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| Summary of Relation properties |
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| (i) R is reflexive if aRa (ii) R is transitive if aRb and bRc implies aRc |
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| (iii) R is symmetric if aRb implies bRa. A relation which is reflexive, transitive, and symmetric is called an equivalence relation. |
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| A relation R defined on the set of integers is defined as R = {(a, b) : a - b is an integer}. |
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| Show that R is an equivalence relation. |
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| (i) For reflexive property: |
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Let   a - a = 0 is an integer  True. |
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| (ii) For symmetric property: |
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a - b is an integer  b - a = -(a - b) is also an integer  True. |
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| (iii) For transitive property: |
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| Let a, b c be integers then (a - b) is an integer, (b - c) is an integer. |
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a - c = (a - b) + (b - c) is also an integer  True. |
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| From (i), (ii) and (iii) R is an equivalence relation. |
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| Show that 'is a brother of' on a set of people is not an equivalence relation. |
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1. For reflexive property: aRa for example, x 'is brother of' himself.  False |
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2. For transitive property : Let a, b, c be three persons. If aRb, bRc then aRc. a is brother of b, and b is brother of c then a is brother of c.  True. |
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3. For symmetric property: If aRb then bRa. Let a be brother of b then b may not be brother of a, because b may be sister of a.  False. |
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| When any one property is not satisfied we say that the relation is not an equivalence relation. |
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| Let A = {members of a family}. The relation R means 'is the father of'. Show that the relation R does not satisfy any property of relation. |
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(i) For reflexive property: aRa  a is the father of a, |
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 a is the father of himself.  False |
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| (ii) For symmetric property : aRb then bRa. |
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If a is the father of b then b is the father of a.  False. |
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| (iii) For transitive property : aRb, bRc then aRc. If a is the father of b and b is the father of c, then a is a father of c. |
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It is not possible as a is the grandfather of c.  False. |
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 None of the properties are satisfied. |
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