Question

Discrete Mathematics.

Let A = {2,4,6,8,10}, and define a relation R on A as ∀x,y ∈ A,xRy ↔ 4|(x−y).

(a) Show R is an equivalence relation.
(b) Give R explicitly in terms of its elements.
(c) Draw the directed graph of R.
(d) List all the distinct equivalence classes of R.

Answer 1

solution: Page 1 * Given a) Reflexive porpesty: Consider x belongs to R, =o, which is an integer, so then h/(x-x) XRx. symmetric property: Consider x,y belongs to R and xRg. Thus x-y is integer. And 9-x==(x-y) is also an integer so y Rs. - consider x,y belongs to R, ary and Then yox and x-y are integers. And Sum of those two expressions will also give some integer. So xR2 . Thus, Ris equivalence relation, ЗRх .

Page-2 b) As per equation, the elements need to be divided by hi So R={(2,6), (2,6), (418), (6,2), (6, 1), (8,4), (10,2), (10,4)} -

Answer 2

solution: Page 1 * Given a) Reflexive porpesty: Consider x belongs to R, =o, which is an integer, so then h/(x-x) XRx. symmetric property: Consider x,y belongs to R and xRg. Thus x-y is integer. And 9-x==(x-y) is also an integer so y Rs. - consider x,y belongs to R, ary and Then yox and x-y are integers. And Sum of those two expressions will also give some integer. So xR2 . Thus, Ris equivalence relation, ЗRх .

Page-2 b) As per equation, the elements need to be divided by hi So R={(2,6), (2,6), (418), (6,2), (6, 1), (8,4), (10,2), (10,4)} -
c)
2 2 4 6 6 8 8 10 10

d ) 2) Distinct equivalence classes of R, So R = {(2,6), (2, 6), (4,8), (6,10), (10,4)} Х Please give me Positive like.