Need help with this proof about set relations

[coasterguy10]

1. I am triying to write a proof that states. Let R be a binary relation on a set A and let R^t be the transitive closure of R. Prove that for all x and y in A, xR^t y if and only if, there is a sequence of elements of A, say X1, X2,..., Xn, such that X= X1, X1RX2,... Xn-1RXn, and Xn = y.

Im not very good with proofs and this one just has me stumped. Any help is appreciated

 

[lippyka]

. Proof: Let x and y be elements of A. First, suppose xR^t y. By definition, this means that there exists a sequence of elements of A (X1, X2, ..., Xn) such that X1RX2, X2RX3, ..., X(n-1)RXn, and Xn = y. Now, for the other direction, suppose there exists a sequence of elements of A (X1, X2, ..., Xn) such that X1RX2, X2RX3, ..., X(n-1)RXn, and Xn = y. Then, by definition, xR^t y. Therefore, we can conclude that xR^t y if and only if there is a sequence of elements of A, say X1, X2,..., Xn, such that X= X1, X1RX2,... Xn-1RXn, and Xn = y.