Question

If A = {1,2} and R = {(1,1), (1,2), (2,1), (2,2)}. Check whether R is an equivalence

relation​

Answer 1

Answer:

yes it is eviqvalance relationship

Answer 2

Answer:

R defined on NxN such that (a,b) R (c,d) → ad=bc

Reflexivity Let $(a,b) \: \epsilon N \times N$

implies (a, b) R (a, b)

∴ R is reflexive on, N×N.

Symmetry Let (a, b), (c, d) ∈N×N,

then (a, b) R (c, d) implies ad = bc

implies cb = da

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

∴ R is symmetric on N×N

Transitivity Let (a,b),(c,d),(e,f),∈N×N.

Then, (a, b) R (c, d) implies ad = bc ... (i)

(c, d) R (e, f) implies cf = de ... (ii)

From Eqs. (i) and (ii), (ad) (cf) = (bc) (de)

implies af = be

implies (a, b) R (e, f)

∴ R is transitive relation on N×N.

∴R is equivalence relations on N×N.

Here reflexive → (1,1),(2,2)

Symmetric→(1,2),(2,1)

Transitive → (1,2),(2,3)

It is prove that R is an equivalence.