CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Show that the relation R on the set N×N 
(a,b)R(c,d) if and only if a+d = b+c is an equivalence relation


Solution

let N={1,2,........,9}
Toolbox:
  • 1.R is an equivalance relation if R is 
  • a) reflexive ie (a,b)∈N×N (a,b)R(a,b)
  • b) symmetric ie (a,b)R(c,d)=>(c,d)R(a,b) (a,b)(c,a)∈N×N
  • c) transitive ie (a,b)R(c,d);(c,d)R(e,f)=>(a,b)R(e,f) (a,b)(c,d),(e,f)∈N×N

A={1,2,3......9}A={1,2,3......9}
 
R in N×N
 
(a,b)R(c,d) if (a,b)(c,d)∈N×N
 
a+b=b+c
 
Consider (a,b)R(a,b)  (a,b)∈N×N
 
a+b=b+a
 
Hence R is reflexive
 
Consider (a,b)R(c,d)given by (a,b)(c,d)∈N×N
 
a+d=b+c=>c+b=d+a
 
=>(c,d)R(a,b)
 
Hence R is symmetric
 
Let (a,b)R(c,d)and(c,d)R(e,f)
 
(a,b),(c,d),(e,f),∈N×N
 
a+b=b+c and c+f=d+e
 
a+b=b+c
 
=>a−c=b−d........(1)
c+f=d+e,,,,,,,,,,,,,,,(2)
 
adding (1) and (2)
 
a−c+c+f=b−d+d+e 
a+f=b+e
 
=>(a,b)R(e,f)
 
R is transitive
 
R is an equivalnce relation
 

flag
 Suggest corrections
thumbs-up
 
0 Upvotes


Similar questions
View More


People also searched for
View More



footer-image