Are the following statements true or false? Justify your answers.

(i) If $x$ and $y$ lie in the same path-component of $X$, then $\Pi_{1}(X, x) \cong \Pi_{1}(X, y)$.

(ii) If $x$ and $y$ are two points of the Klein bottle $K$, and $u$ and $v$ are two paths from $x$ to $y$, then $u$ and $v$ induce the same isomorphism from $\Pi_{1}(K, x)$ to $\Pi_{1}(K, y)$.

(iii) $\Pi_{1}(X \times Y,(x, y))$ is isomorphic to $\Pi_{1}(X, x) \times \Pi_{1}(Y, y)$ for any two spaces $X$ and $Y$.

(iv) If $X$ and $Y$ are connected polyhedra and $H_{1}(X) \cong H_{1}(Y)$, then $\Pi_{1}(X) \cong \Pi_{1}(Y)$.