Let f A rightarrow B and let X Y A and U V B Prove that a fx

Let f: A rightarrow B, and let X, Y A and U, V B. Prove that (a) f(x) U if and only if X f^-1(U).

Solution

Let f:AB, and let XA and U B. Prove that f(X) U if and only if Xf^-1(U). Let p be an arbitrary element of X . Since f: AB, therefore f(p) B. If f(X) U, then f(p) U so that p f^-1(U). Hence X f^-1(U). Now let X f^-1(U) and let p be an arbitrary element of X . Then p f^-1(U) so that u U such that f(p)= u. This shows that f(X) U.