Give proofs or counter-example for the following statements:
i)If X and Y are path connected subsets of Z and X/Y ( X intersection Y)
is non empty then X/Y is path connected.
ii) If X and Y are path connected subsets of Z the X/Y(union) is
path-connected.
iii)If X and Y are path connected subsets of Z and X/Y ( X intersection Y)
is non empty then X/Y(union) is path connected.
This solution is comprised of a detailed explanation to give proofs or counter-example for the following statements.