1Questions and Answers

Given two sets $X$ and $Y$, a function $f$ from $X$ to $Y$ is a rule that selects for each element of $X$ exactly one element of $Y$. The mathematical notation $f:X\rightarrow Y$ says that $f$ is a function from $X$ to $Y$. The element $y\in Y$ that is selected for $x\in X$ is called the value of the argument $x$ and is written as $f(x)$. Note that $X,Y,x,y$ are variables here that are placeholders for concrete sets and concrete elements of these sets.

For example, if $X$ is the set of countries in the world and $Y$ is the set of cities in the world, then we can consider the function $f:X\rightarrow Y$ that selects for each country its capital. The value of the argument Austria is Vienna,

Because sets are mathematical entities and do not correspond to bags in real life, they do not actually contain countries or cities, but just text symbols that identify the countries and cities.

We can also think of $X$ as questions in a quiz, and $Y$ as answers. Then a function $f$ provides an answer $y=f(x)\in Y$ for every question $x\in X$. Our example function above provides the answer to every question of the form What is the capital of country $x$?

In this example, there are elements of $Y$ (for example, Zurich) that are not selected for any $x$, so there may be answers that will never be given. That’s ok. The only thing we require is that for every question, there is an answer. We also allow that several questions have the same answer. As an example, consider the function corresponding to the question What is the first letter of country $x$’s name? This gives the same answer A for Austria, or Australia, or Albania, or....

The mathematician creates another theorem out of this.