6. Let A = {-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6) and define the relation R as: m R n (m n). R is an equivalence relation. List the distinct equivalence classes of R.

Answer with explanation:

⇒A = {-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}. A subset of A * A is called Relation on R.

→We have to find an Equivalence relation on R. A Relation is said to be Equivalence if it is (a) Reflexive (b) Symmetry (c) Transitive.

A relation R is said to be reflexive if, (a, a) ∈R.

A relation R is said to be Symmetric if, (a, b) ∈R, (b, a) ∈R.

A relation R is said to be Transitive if, (a, b) ∈R, (b, c) ∈R⇒ (a, c) ∈R.

→There can be many equivalence Relation on R. For example

⇒R

=A*A

={ (-6,-6), (-5,-5), ... (0,0), (1,1), ... (5,5), (6,6), (-6,-5), (-5,-6), (-6,-4), (-4,-6), (-6,-3), (-3,-6), ... (-6,6), (6,-6), (-5,-4), (-4,-5), (-5,-3), (-3,-5), ... (-5,6), (6,-5), ... (-4,-3), (-3,-4), (-4,-2), (-2,-4) ... (-4,6), (6,-4), (0,1), (1,0), (0,2), (2,0), ... (0,6), (6,0), (1,2), (2,1), (1,3), (3,1), (1,4), (4,1), (1,5), (5,1), (1,6), (6,1), (2,3), (3,2), ... (2,6), (6,2), (3,4), (4,3), ... (3,6), (6,3), (4,5), (5,4), (4,6), (6,4), (5,6), (6,5) }

Equivalence class of R for an equivalence relation:

The set of distinct elements of A which is related to elements of A.

-6 = {-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

-5 = {-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

-4={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

-3={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

-2={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

-1={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

0={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

1={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

2={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

3={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

4={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

5={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}

6={-6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6}