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lemonthree

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S is reflexive

S is symmetric

S is transitive

S is reflexive and symmetric

S is reflexive and transitive

I assume that S acts on a set A. So let a,b,c be elements of A.

For S to be reflexive, for all a in A, a S a.

For S to be symmetric, for all a,b in A, if a S b, then b S a.

For S to be transitive, for all a,b,c in A, if a S b and b S c, then a S c.

Now I got to compose S with S, and I know that the standard definition of R∘S = the relation of A to C, where (a,c) is an element of A x C such that there exists b in B s.t. (a,b) in R and (b,c) in S.

So I have to adapt this standard definition of R∘S to S∘S? I assume that S∘S is the relation of A to A, where (a,c) is an element of A x A such that there exists b in A s.t. (a,b) in S and (b,c) in S.

Therefore, I think S∘S=S implies S is reflexive, symmetric and transitive?

1. For all a in A, a S a.

2. For all a,b in A, if a S b, then b S a. Likewise, if b S a, then a S b.

3. For all a,b,c in A, if a S b and b S c, then a S c. Likewise, if c S b and b S a, then c S a.

So with that, S should be reflexive and symmetric / reflexive and transitive too? Have I misunderstood any parts?