Use Discrete Math Please:

iA A A is the identity function on a set A, i.e., iA(x) = x

for all x element of A. Let f A B. A left inverse of f is a function g B A such that g X f = A. A right inverse of f is a function h B A such that f X h = B.

Thus an inverse of f, as de ned on page 147 of the text, is a function g B A that is both a left inverse and a right inverse of f. A function is left-invertible if it has a left inverse, right-invertible if it has a right inverse, and invertible if it has an inverse. 48. Prove: A function f A B has a left inverse if and only if it is one-to-one. 49. Prove: A function f A B has a right inverse if and only if it is onto. 50. (a) Let f A B and g; h B A. Prove: If g is a left inverse of f and h is a right inverse of f, then g = h. (b) Use (a) to prove that a function f A B has at most one inverse. 1