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 R^{2} 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 R^{2}:

(3,1) in R and (1,2) in R, whence (3,2) in R^{2}.

On the other hand, (2,3) does not belongs to R^{2}.

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 R^{2}.

Then R^{2} can be represented via the following table:

1 2 3 4 5

1 * * * * *

2 * * * *

3 * * * * *

4 * * * *

5 * * * * *

Similar calculations shows that

Then R^{3}=R^{4}=R^{5} = X x X:

1 2 3 4 5

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 R

For instance let us check whether (3,2) in R

(3,1) in R and (1,2) in R, whence (3,2) in R

On the other hand, (2,3) does not belongs to R

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 R

Then R

1 2 3 4 5

1 * * * * *

2 * * * *

3 * * * * *

4 * * * *

5 * * * * *

Similar calculations shows that

Then R

1 2 3 4 5

1 * * * * *

2 * * * * *

3 * * * * *

4 * * * * *

5 * * * * *

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