Question

If the co-domain does not contain all the elements of range , can it be a function?
like co-domain {3,4,5,6,7}
range {3,4 11}. normally we know range is a subset of co-domain. but here it does not . so is it possible?

Answer 1

Answer on Question #51704 – Math – Real Analysis

Question. If the co-domain does not contain all the elements of range, can it be a function, like co-domain $\{3,4,5,6,7\}$ range $\{3,4,7\}$? Normally we know range is a subset of co-domain. But here it does not. So is it possible?

Answer. By definition, a function $f:X\to Y$ is the correspondence which associates to each $x\in X$ a unique element from $Y$ denoted by $f(x)$.

The set $X$ is then called the domain of $f$, the set $Y$ is said to be the co-domain, and the set $f(X)=\{f(x)\in Y\mid x\in X\}$ is the range of $f$.

The above definition does not require that the range $f(X)$ coincides with all the co-domain $Y$. So in general, the range of the function can be a proper subset of the co-domain:

$f(X)\subsetneq Y.$

The functions for which the range coincides with the co-domain $f(X)=Y$ are called surjective.

For example, the functions $\sin,\cos:\mathbb{R}\to\mathbb{R}$ are not surjective have the same range $[-1,1]$.

Also notice that every function $f:X\to Y$ induces a surjective function $\hat{f}:X\to f(X)$ defined by $\hat{f}(x)=f(x)$. In other words, we can always “replace” co-domain with the range to get a surjective fucntion.

Summarize all that is said above: in general, the range of the function can be a proper subset of the co-domain.