Linear Optimization Practice problem

3. Given a nonempty polyhedron P={(x,y)RnRk:Ax+Byb}, let Q denote its projection onto x-space, i.e., Q={xRn:yRk,Ax+Byb}. Prove or disprove the following statements by counterexamples: 1) Suppose that (x^,y^) is an extreme point of P. Is x^ an extreme point of Q ? 2) Suppose that x^ is an extreme point of Q. Does there exist a y^ such that (x^,y^) is an extreme point of P ? 3) Suppose that x^ is an extreme point of Q and P does not contain a line. Does there exist a y^ such that (x^,y^) is an extreme point of P ? 3. Given a nonempty polyhedron P={(x,y)RnRk:Ax+Byb}, let Q denote its projection onto x-space, i.e., Q={xRn:yRk,Ax+Byb}. Prove or disprove the following statements by counterexamples: 1) Suppose that (x^,y^) is an extreme point of P. Is x^ an extreme point of Q ? 2) Suppose that x^ is an extreme point of Q. Does there exist a y^ such that (x^,y^) is an extreme point of P ? 3) Suppose that x^ is an extreme point of Q and P does not contain a line. Does there exist a y^ such that (x^,y^) is an extreme point of P