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}

ratio is 2 : 9

the answer is three

13.5 : 3