Given a mapping (function) f from A to f(A):

Definition: f is injective if

1) x1=x2 -> f(x1)=f(x2) Ex: sqrt(4)=+2, sqrt(4)=-2

2) X1≠x2 -> f(x1)≠f(x2) Ex: 2^{2}=(4), (-2)^{2}=4

1) and 2) imply the alternate definition:

3) f(x1)=f(x2) -> x1=x2

4) f(x1)≠f(x2) -> x1≠x2

1) & 4) are equivalent.

2) & 3) are equivalent.

Any of the combinations (tests) 1),2); 1),3); 4),2); 4),3) establish an injection.

Surjective is relative:

If B=f(A), f:A->B is surjective. (if f is also injective, called bijective, or 1-1 onto,)

If B=f(A) is a subset of C, f:A->C is not surjective. (if f is injective, called 1-1 into,)