Question

We say a function f : A B is surjective (onto) iff for all ye B, EA exists so that f() = y. What does it mean for a function not to be surjective?

Answer 1

Answer:

We say that a function $f:A\rightarrow B$ is surjective or onto if and only if for every element $b\in B$ there exist an element $a\in A$, such that $b=f(a)$.

This definition says that a function is surjective if every element of $B$ has a pre-image, or that the image of $A$ by $f$ ‘‘fills’’ $B$ completely.

So, if the function $f$ is not surjective, there is, at least, one element of $B$ without pre-image. In other words, that we a have a proper inclusion $f(A)\subset B$, and $f(A)\neq B$ .

Step-by-step explanation: