Question

if f:X to Y and g:Y to Z then the domain of gof is X and co-domain is Z , what is the domain and co-domain of fog, fof, gog, here in this case?

Answer 1

Answer on Question #51628 – Math – Functional Analysis

Question. If $f:X\to Y$ and $g:Y\to Z$ then the domain of $g\circ f$ is $X$ and co-domain is $Z$. What is the domain and co-domain of $f\circ g$, $f\circ f$, $g\circ g$, here in this case?

Solution. Notice that the composition of map $g\circ f$ is possible only when the codomain (the image) of $f$ coincides with the domain of $g$. In other words, we could write the composition of arrows:

$g\circ f:X\xrightarrow{f}Y\xrightarrow{g}Z.$

However, in general, neither of the compositions $f\circ g$, $f\circ f$, $g\circ g$ are possible.

Nevertheless, if $f\circ g$ is defined, then we should have that $Z=X$:

$f\circ g:Y\xrightarrow{g}Z\ =\ X\xrightarrow{f}Y.$

In this case $Y$ is the domain and co-domain of $f\circ g$.

Similarly, if $f\circ f$ is defined, then we should have that $X=Y$:

$f\circ f:X\xrightarrow{f}Y\ =\ X\xrightarrow{f}Y.$

In this case $X=Y$ is the domain and co-domain of $f\circ f$.

By the same reason, if $g\circ g$ is defined, then we should have that $Y=Z$:

$g\circ g:Y\xrightarrow{g}Z\ =\ Y\xrightarrow{g}Z.$

In this case $Y=Z$ is the domain and co-domain of $f\circ f$.

Answer.

1) The composition $f\circ g$ is defined only for $Z=X$, and in this case $Y$ is the domain and co-domain of $f\circ g$.

2) The composition $f\circ f$ is defined only for $X=Y$, and in this case $X=Y$ is the domain and co-domain of $f\circ f$.

3) The composition $g\circ g$ is defined only for $Y=Z$, and in this case $Y=Z$ is the domain and co-domain of $g\circ g$.

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