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Answer to Question #2558 in Discrete Mathematics for zaara

Question #2558
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
Expert's answer
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 * * * * *

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