What does "$\max\{m,n\}$" mean (eg, when $m$ and $n$ are degrees of polynomials)?

I was reading a mathematics book, and I found if $f(X)$ is a polynomial of degree $n$, and $g(X)$ is a polynomial of degree $m$, then $f(X)\pm g(X)$ is a polynomial of degree less than or equal to $\max\{m,n\}$.

What does the notation "$\max\{m,n\}$" mean?

Solution 1:

Short answer: $\max\{m,\,n\}$ is the maximum of $m$ and $n$, i.e. $m$ if $m\ge n$ or $n$ otherwise.

Long answer: In general $\max S$ is the maximum element of the set $S$, if that maximum exists. A $2$-element set of polynomial degrees (which are non-negative integers or, in the case of the zero polynomial, $-\infty$ in some conventions) denoted $\{m,\,n\}$ certainly has such a maximum; indeed, any finite set of extended real numbers does. So $\max\{m,\,n\}$ is $m$ if $m\ge n$, or $n$ otherwise.