A,B,C are sets and f: A-->B and g: B-->C. Prove:
1)If f and g are one-to-one, then so is g o f.
2)If g o f is one-to-one, then g need not be one-to-one.
Any advice?
Of course you know what to prove
gof(x) = gof(y) x = y
Since g(f(x)) = g(f(y)) and g is one-one, we have f(x) = f(y). Since f is also 1-1, we have x=y.
2) is nice, I will give you a general hint,try it
General Hint: Draw a few blobs(actually 3, they are the sets). Mark some points in them(they are the elements). Now map points from one blob to another, remembering they conditions in data. Try your best to prove the question wrong. You will see why it must be right. Geometric intuition is your best pal, learn to use him.