description

[calculatingsomething]

show, that the description f: X :arrow: Y is an injection if and only if f^-1(f(B)) = B on every B C X


I think it will go:

X Y

1 :arrow: A
B
2 :arrow: C

3 :arrow: D

But I'm not sure about this.

 

[chrisr]

A function is an injection, if for f(x) = f(y), then x = y,
also if f(x) is not = f(y), then x is not = y.

An injection may be referred to as one-to-one as it maps a distinct element to a distinct element.

To show that a function is not injective, you may find x[sub:255kmcjl]1[/sub:255kmcjl] different to x[sub:255kmcjl]2[/sub:255kmcjl], for which f(x[sub:255kmcjl]1[/sub:255kmcjl])=f(x[sub:255kmcjl]2[/sub:255kmcjl]),
as in the case of quadratics in x.
A graph, then is not injective if f(A) = f(B) where B is different to A.
Hence, f(x[sub:255kmcjl]1[/sub:255kmcjl]) = B means only f[sup:255kmcjl]-1[/sup:255kmcjl](B) = x[sub:255kmcjl]1[/sub:255kmcjl] for the function to be injective,
for all B that are elements of the domain of x.