Unwrapping Definitions

This is another form of garbage management. Here, we provide tips for understanding definitions, rather than reducing problems to something doable.

To understand a definition, I have two tips: 1) do it piece by piece, 2) try to look at a simpler version of it, with concrete values/objects.

Continue to see an example.

For example, here is a the mathematical definition of a continuous function:

A function $f:\mathbb{R}\to\mathbb{R}$ is continuous if and only if $\forall \epsilon>0$ $\exists \delta>0$ such that $\forall x\in\mathbb{R}$, $|x-c|<\delta\implies |f(x)-f(c)|<\epsilon$.

First of all, what is $\forall$, $\exists$ and $\implies$? These symbols mean "for all" ($\forall$), "there exists" ($\exists$) and "implies" ($\implies$).

So the definition says, for any $\epsilon>0$, so say for $\epsilon=1$, we can find a positive $\delta$ such that $|f(x)-f(c)|<1$ whenever $|x-c|<\delta$.

What sorts of $x$ satisfy the criterium $|x-c|<\delta$? These are the $x$'s in the range $(c-\delta,c+\delta)$, so these are close to $c$. Thus, "$|f(x)-f(c)|<1$ whenever $|x-c|<\delta$" means that $f(x)$ is within $1$ distance of $f(c)$ whenever $x$ is close to $c$.

So we're basically done with the decryption: $f$ is continuous means that $f(x)$ gets close to $f(c)$ whenever $x$ is close enough to $c$.

For a set $S\subset \mathbb{R}$ with $x\le K$ $\forall x\in S$ for some $K$, the supremum of $S$, or $\sup S$, is defined as a quantity $M$ such that $\forall \epsilon>0$ $\exists x\in S$ $x>M-\epsilon$, but $\forall x\in S$ $x\le M$.

Try to rephrase the definition of the supremum less incomprehensibly.

Read this problem from the 2000 China National Olympiad:

A test contains $5$ multiple choice questions which have $4$ options in each. Suppose each examinee chose one option for each question. There exists a number $n$, such that for any $n$ sheets among $2000$ sheets of answer papers, there are $4$ sheets of answer papers such that any two of them have at most $3$ questions with the same answers. Find the minimum value of $n$.

Try to understand what this problem wants. After that, show this problem to a friend and test yourself by trying to explain it to her/him.