Question

Let `R` be a relation defined on the set of natural numbers `N` as `R={(x , y): x , y in N , 2x+y=41}` Find the domain and range of `R` . Also, verify whether `R` is (i) reflexive, (ii) symmetric (iii) transitive.

Answer 1

`2x+y = 41`
`=>x = (41-y)/2`
`:. y in {1,3,5,7....39}`, which is range of `R`.
`:. x in {1,2,3,4...19,10}`, which is domain of `R`.
Now, `R = {(x,y): x,y in N and 2x+y = 41}`
Now, for `(a,a)`,
`2a+a = 41, => a = 41/3 => a !in N`
`:. (a,a) !in R`
`:. R` is not reflexive.

`(a,b) in R`.
It means `2a+b = 41`.
But, it is not neccessaty that `2b+a = 41`. `:. (b,a) notin R`.
`:. R` is not symmetric.

If `(a,b) in R`,
Then, `2a+b = 41`.
If `(b,c) in R`,
Then, `2b+c = 41`
Then, `2a+b+2b+c = 82`
`=>2a+3b+c =82`
For `2a+c = 41`, `3b = 41 or b = 41/3 `, which is not true as `b in N`.
`:. (a,c) notin R`.
`:. R` is not transitive.