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# Determine whether the following relations are reflexive, symmetric or transitive :

(i) R on Z defined by (a, b)R|a-b|5.

(ii)R on Q - {0} defined by (a, b)Rab = 4.

# If R is a relation in **NN defined by (a, b) R (c, d) if and only if a+d = b+c, show that R is an equivalence relation.

# Let N denote the set of all natural numbers and R be the relation on NN defined by (a, b) R (c, d)ad(b+c) = bc(a+d). Show that R is an equivalence relation on N*N**.

# If and are two equivalence relation on a set A, then prove that is also an equivalence relation on A.

# Show that the number of equivalence relations on the set {1,2,3} containing (1,2) and (2,1) is two.

Asked by abhinavsaini123 | 29th May, 2015, 07:55: PM

Expert Answer:

Please ask one query in a single post. The first two questions has been explained in other posts. Please find the explanation for the 3rd question here:

R be the relation on **N*N** defined by (a, b) R (c, d)ad(b+c) = bc(a+d).

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

$T r a n s i t i v i t y colon I f space open parentheses a comma b close parentheses R open parentheses c comma d close parentheses rightwards double arrow a d open parentheses b plus c close parentheses equals b c open parentheses a plus d close parentheses.. left parenthesis 1 right parenthesis I f space open parentheses c comma d close parentheses R open parentheses e comma f close parentheses rightwards double arrow c f open parentheses d plus e close parentheses equals d e open parentheses c plus f close parentheses.. left parenthesis 2 right parenthesis F r o m space left parenthesis 1 right parenthesis space fraction numerator b plus c over denominator b c end fraction equals fraction numerator a plus d over denominator a d end fraction rightwards double arrow 1 over c plus 1 over b equals 1 over d plus 1 over a.... left parenthesis 3 right parenthesis F r o m space left parenthesis 2 right parenthesis space fraction numerator d plus e over denominator d e end fraction equals fraction numerator c plus f over denominator c f end fraction rightwards double arrow 1 over e plus 1 over d equals 1 over f plus 1 over c... left parenthesis 4 right parenthesis A d d i n g space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis space comma space w e space g e t 1 over c plus 1 over b plus 1 over e plus 1 over d equals 1 over d plus 1 over a plus 1 over f plus 1 over c rightwards double arrow 1 over b plus 1 over e equals 1 over a plus 1 over f rightwards double arrow a f open parentheses b plus e close parentheses equals b e open parentheses a plus 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 r e l a t i o n space i s space t r a n s i t i v e. space S o 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. space H e n c 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.$

Answered by satyajit samal | 31st May, 2015, 04:54: PM

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