Suppose A, B and C are sets and f: A->B, and g: B->C, prove:
if f and g are one-to-one, so is (g o f),
If f and g are onto so is (g o f),
and if f and g are bijections, so is (g o f)
Now, our book defines one-to-one like so..
a function f is one-to-one whenever \(\displaystyle (x,b) \epsilon f\), \(\displaystyle (y,b) \epsilon f\), we must have \(\displaystyle x = y\).
Which makes sense, if you have to values in the im f, and they are the same, then the values that correspond to them in the domain must be the same as well. I'm just not sure how to start proving these, maybe someone can push me in the right direction?
if f and g are one-to-one, so is (g o f),
If f and g are onto so is (g o f),
and if f and g are bijections, so is (g o f)
Now, our book defines one-to-one like so..
a function f is one-to-one whenever \(\displaystyle (x,b) \epsilon f\), \(\displaystyle (y,b) \epsilon f\), we must have \(\displaystyle x = y\).
Which makes sense, if you have to values in the im f, and they are the same, then the values that correspond to them in the domain must be the same as well. I'm just not sure how to start proving these, maybe someone can push me in the right direction?
