Question

Let N be the set of all natural numbers and let R be a relation on `N×N` , defined by `(a , b)R(c , d) iff a d=b c` for all `(a , b),(c , d) in N × Ndot` . Show that `R` is an equivalence relation on `N × N` .

Answer 1

Here, `(a,b)R(c,d) <=> ad = bc` for all `(a,b),(c,d) in N xx N.`
First we will check `R` for reflexive.
For, `(a,b)R(a,b) `,
`=>ab = ba`, which is true.
So, `R` is reflexive.
Now, we will check for symmetric.
For, `(a,b)R(c,d)`,
`=>ad = bc`
`=>bc = ad`
`=>cb = da`
`=>(c,d) R(a,b) ` is true.
So, `R` is symmetric.
Now, we will check `R` fo transtivity.
For, `(a,b)R(c,d) and (c,d)R(e,f)`
`=> ad = bc and cf = de`
`=> a/b = c/d and c/d = e/f`
`=> a/b = e/f`
`=> af = be`
So, `(a,b)R(e,f)` is true.
`:. R` is transitive.
As `R` is reflexive, symmetric and transitive, `R` is an equivalence relation.