0
$\begingroup$

Say we have three disjoint sets $A$, $B$ and $C$. And three functions $f$, $g$ and $h$ where $f:A \rightarrow B$, $g:A\rightarrow C$ and $h$ takes the inputs $h(f(a_1),g(a_2))$ for $a_1, a_2 \in A$ .

What is the domain of $h$?

I'm reading Charles C. Pinter's a Book of Abstract Algebra and drew up a little problem based on the second chapter on operations. I asked myself whether $h$ would be a binary operation or not, assuming its range is $A$. But I'm feeling unsure as to its domain. I've searched for how to determine domains of multi-variate functions and from that I guess that the answer would be some subset of $B \times C$ since those are the ranges of $f$ and $g$ respectively.

But, just looking at the function $h(f(a_1),g(a_2))$, for any two $a_n$ I'd give, I would get a valid answer. So it 'feels' as if $A \times A$ is the domain of $h$ and thus $h$ would be a binary operation.

$\endgroup$
0
$\begingroup$

I am a bit embarrassed. Took a walk and the solution fell on me.

What I failed to see was that the function $h(f(a_1),g(a_2))$ is a composite function: $h\circ f\times g$ and since the domain of both $h$ and $f$ is $A$ it follows that the domain for $h$ is $A\times A$. Since $h: A\times A \rightarrow A$, $h$ is a binary operation.

Another point of confusion was that I mentally interchanged $h(b,c)$ for $b\in B$ and $c\in C$ with $h(f(a_1),g(a_2))$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.