Question

Show that the relation R on the set N*N
defined by (a,b) R (c,d) defined by the
defined by a d = b c is an equivalence relation

Answer 1

SOLUTION

TO PROVE

The relation R on the set N × N defined by (a,b) R (c,d) if and only if ad = bc is an equivalence relation

PROOF

Here a relation R on the set N × N defined by (a,b) R (c,d) iff ad = bc

CHECKING FOR REFLEXIVE

Let (a, b) ∈ N × N

∵ ab = ab

∴ (a,b) R (a, b)

∴ R is Reflexive

CHECKING FOR SYMMETRIC

Let (a, b) & ( c, d) ∈ N × N

Suppose that (a,b) R (c,d)

$\implies \sf{ad = bc}$

$\implies \sf{cb = da}$

∴ (c,d) R (a, b)

∴ (a,b) R (c,d) implies (c,d) R (a, b)

∴ R is Symmetric

CHECKING FOR TRANSITIVE

Let (a, b) , ( c, d) & (x, y) ∈ N × N

Suppose that (a,b) R (c,d) & (c, d) R (x, y)

Now (a,b) R (c,d) gives ad = bc

Again (c, d) R (x, y) gives cy = dx

On multiplication we get

$\sf{adcy = bcdx}$

$\implies \sf{ay = bx}$

∴ (a,b) R (x, y)

∴ R is Transitive

Hence R is an Equivalence Relation

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