1. Determine whether the following relations are reflexive, symmetric or transitive :
  • R on Z defined by (a, b)element ofRleft right double arrow|a-b|less or equal than5.
  • R on Q - {0} defined by (a, b)element ofRleft right double arrowab = 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)left right double arrowad(b+c) = bc(a+d). Show that R is an equivalence relation on N*N.

4. If R subscript 1 and R subscript 2 are two equivalence relation on a set A, then prove that R subscript 1 intersection R subscript 2 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.

Asked by abhinavsaini123 | 29th May, 2015, 12:56: PM

Expert Answer:

Please ask one query in a single post. The explanation for the first question is given below:
P a r t space 1 colon space R space d e f i n e d space b y space open parentheses a comma b close parentheses element of R left right double arrow open vertical bar a minus b close vertical bar less or equal than 5 R e f l e x i v e colon space F o r space a n y space open parentheses a comma a close parentheses element of R comma space open vertical bar a minus a close vertical bar equals 0 less or equal than 5. space H e 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.  S y m m e t r i c colon space F o r space a n y space open parentheses a comma b close parentheses element of R comma space space open vertical bar a minus b close vertical bar less or equal than 5 rightwards double arrow open vertical bar b minus a close vertical bar less or equal than 5 rightwards double arrow open parentheses b comma a close parentheses element of R therefore space open parentheses a comma b close parentheses element of R rightwards double arrow open parentheses b comma a close parentheses element of R. space H e n c e comma space t h e 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 open parentheses 1 comma 2 close parentheses element of R comma space open parentheses 2 comma space 7 close parentheses element of R comma space b u t space open parentheses 1 comma 7 close parentheses not an element of R space open parentheses because open vertical bar 1 minus 7 close vertical bar equals 6 space w h i c h space i s space m o r e space t h a n space 5. 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 n o t space t r a n s i t i v e.  P a r t space 2 colon space R space o n space Q minus left curly bracket 0 right curly bracket space d e f i n e d space b y space open parentheses a comma b close parentheses element of R left right double arrow a b equals 4 R e f l e x i v e colon space F o r space a n y space apostrophe a apostrophe space i n space Q comma space v e r i f y space i f space left parenthesis a comma a right parenthesis space b e l o n g s space t o space R space o r space n o t. 1 half element of Q comma space b u t space open parentheses 1 half comma 1 half close parentheses not an element of R. space H e n c e comma space t h e space r e l a t i o n space i s space n o t space r e f l e x i v e.  S y m m e t r i c colon space I f space left parenthesis a comma b right parenthesis element of R comma space t h e n space a b equals 4 rightwards double arrow b a equals 4 rightwards double arrow open parentheses b comma a close parentheses element of R. H e n c e comma space t h e 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 left parenthesis 1 comma 4 right parenthesis element of R comma space open parentheses 4 comma 1 close parentheses element of R comma space b u t space open parentheses 1 comma 1 close parentheses not an element of R. 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 n o t space t r a n s i t i v e.

Answered by satyajit samal | 30th May, 2015, 11:15: PM