Respuesta :
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}
