1. Determine whether the following relations are reflexive, symmetric or transitive : R on Z defined by (a, b)R|a-b|5. R on Q - {0} defined by (a, b)Rab = 4. 2. If R is a relation in N*N defined by (a, b) R (c, d) if and only if a+d = b+c, show that R is an equivalence relation. 3. Let N denote the set of all natural numbers and R be the relation on N*N defined by (a, b) R (c, d)ad(b+c) = bc(a+d). Show that R is an equivalence relation on N*N. 4. If and are two equivalence relation on a set A, then prove that is also an equivalence relation on A. 5. Show that the number of equivalence relations on the set {1,2,3} containing (1,2) and (2,1) is two.

R is a relation in N*N defined by (a, b) R (c, d) if and only if a+d = b+c,

$R e f l e x i v i t y colon space F o r space a n y space left parenthesis a comma space b right parenthesis element of N asterisk times N space comma space F o r space left parenthesis a comma b right parenthesis R left parenthesis a comma b right parenthesis space t o space b e space t r u e comma space a plus b space s h o u l d space b e space e q u a l space t o space b plus a comma space w h i c h space i s space a l w a y s space t r u e. H e n c e comma space R space i s space r e f l e x i v e. S y m m m e t r i c colon space I f space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses rightwards double arrow a plus d equals b plus c rightwards double arrow c plus b equals d plus a rightwards double arrow open parentheses c comma d close parentheses R open parentheses a comma b close parentheses H e n c e comma space space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses rightwards double arrow open parentheses c comma d close parentheses R open parentheses a comma b close parentheses H e n c e comma space t h e space g i v e n space r e l a t i o n space i s space s y m m e t r i c. T r a n s i t i v e colon space space I f space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses rightwards double arrow a plus d equals b plus c.. left parenthesis 1 right parenthesis space I f space open parentheses c comma d close parentheses R open parentheses e comma f close parentheses rightwards double arrow c plus f equals d plus e... left parenthesis 2 right parenthesis A d d i n g space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis thin space comma space w e space g e t a plus d plus c plus f equals b plus c plus d plus e rightwards double arrow a plus f equals b plus e rightwards double arrow open parentheses a comma b close parentheses R open parentheses e comma f close parentheses H e n c e comma space space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses space a n d space open parentheses c comma d close parentheses R open parentheses e comma f close parentheses rightwards double arrow open parentheses a comma b close parentheses R open parentheses e comma f close parentheses H e n c e comma space t h e space g i v e n space r e l a t i o n space i s space t r a n s i t i v e. S i n c e comma space t h e space r e l a t i o n space i s space r e f l e x i v e comma space s y m m e t r i c space a n d space t r a n s i t i v e comma space i t space i s space a n space e q u i v a l e n c e space r e l a t i o n.$

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Introduction to Relations

Concept of relations and types of relation

Introduction To Relations

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CBSE Class 12-science Answered

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Introduction to Relations

Concept of relations and types of relation

Introduction To Relations

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