Let E and E be countable sets, and F : E E a mapping (function). Let X0,X1,... be a Markov chain with state space E and transition matrix .

Suppose that F has the property that whenever a pair of elements (e1,e2) in E satisfies F (e1) = F (e2), then (e1, y) = (e2, y) for all y E. That is, the rows of corresponding to e1 and e2 coincide. Under this condition on F, show that F(X0),F(X1),... is a Markov chain. What is the transition matrix for this Markov chain on E ?