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Am I right in the following?

Let $$A = {1, 2, 3, 4, 5}$$ and consider the following relation on A: $$R = {(1, 2),(2, 3),(3, 4),(4, 5),(5, 1)}$$

a)

Here I am to find the composition of R on R. I got this:

$$R^2={(1,3),(2,4),(3,5),(4,1)}$$

b)

$$R={(1,2),(2,3),(3,4),(4,5),(5,1)}$$

The transitive closure is

$$R={(1,2),(2,3),(1,3)(3,4),(4,5),(4,1),(3,5),(5,1)}$$

EDIT1: I added (4,1), because we have (4,5),(5,1), which would be (a,b),(b,c) and the transitive closure of those would be (4,1).

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Not exactly.

a) You missed $(5,2)$, otherwise it's ok.

b) Try to prove that all pairs are in the transitive closure.

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