Question

Let R be the relation on the set {1,2,3,4,5} containing the order pairs&
(1,1) ,(1,2) (1,3) ,(2,3) (2,4) ,(3,1) (3,4) ,(3,5) ,(4,2) (4,5) ,(5,1) (5,2) ,and (5,4)
Find
1-R^3
2-R^4
3-R^5

Answer 1

Let X = {1,2,3,4,5}.
Relation R can be represented via the following table:
1 2 3 4 5
1 * * *
2 * *
3 * * *
4 * *
5 * * *

Then R² consists of all pairs (x,z) such that there exists y in X such that (x,y)in R and (yz)in R

For instance let us check whether (3,2) in R²:
(3,1) in R and (1,2) in R, whence (3,2) in R².

On the other hand, (2,3) does not belongs to R².
Indeed, only (2,3) and (2,4) belong to R, but (3,2) and (4,2) do not.

Similarly we can check all pairs (x,y) to belongs to R².
Then R² can be represented via the following table:
1 2 3 4 5
1 * * * * *
2 * * * *
3 * * * * *
4 * * * *
5 * * * * *

Similar calculations shows that
Then R³=R⁴=R⁵ = X x X:
1 2 3 4 5
1 * * * * *
2 * * * * *
3 * * * * *
4 * * * * *
5 * * * * *