Let $X_{0}, X_{1}, X_{2}, \ldots$ be independent identically distributed random variables with $\mathbb{P}\left(X_{i}=1\right)=1-\mathbb{P}\left(X_{i}=0\right)=p, 0<p<1$. Let $Z_{n}=X_{n-1}+c X_{n}, n=1,2, \ldots$, where $c$ is a constant. For each of the following cases, determine whether or not $\left(Z_{n}: n \geqslant 1\right)$ is a Markov chain: (a) $c=0$; (b) $c=1$; (c) $c=2$.

In each case, if $\left(Z_{n}: n \geqslant 1\right)$ is a Markov chain, explain why, and give its state space and transition matrix; if it is not a Markov chain, give an example to demonstrate that it is not.