Let `A={1,\ 2,\ 3,\ ,\ 9}` and `R` be the relation on `AxxA` defined by `(a ,\ b)R\ (c ,\ d)` if `a+d=b+c` for all `(a ,\ b),\ (c ,\ d) in AxxA` . Prove that `R` is an equivalence relation and also obtain the equivalence class [(2, 5)].

Answer

N/a

Explanation

Given that , `A={1,2,3, …, 9} `and `(a,b),R(c,d)` if `a+d=b+c` for `(a,b) in A xx A` and `(c,d) in A xx A.`
Let `(a,b) R (a,b)`
`implies a+b=b+a, AA a,b in A`
which is true for any `a, b in A.`
Hence, R is reflexive.
Let `(a,b) R (c,d) " " a+d=b+c`
`c+b=d+a implies (c,d) R (a,b)`
So, R is symmetric.
Let `(a,b) R (c,d)` and `(c,d) R (e,f)`
` a+d=b+c` and `c+f=d+e`
`a+d=b+c`and `d+ e=c+f`
`(a+d)-(d+e)=(b+c)-(c+f)`
`(a-e)=b-f`
`a+f=b+e`
`(a,b) R (e,f)`
So, R is transitive.
Hence, R is an equivalence relation.
Now, equivalence class containing `[(2,5)]` is `{(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)} `.