Set theory Examples

Basic operations

Let $A=\{1,2,3\}$ and $B=\{3,4\}$. Then

• $A\cup B=\{1,2,3,4\}$
• $A\cap B=\{3\}$
• $A\setminus B=\{1,2\}$
• $A\times B=\{(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)\}$

The Cartesian product is where relations and functions live.

A relation that is not a function

Let $R=\{(1,a),(1,b),(2,b)\}\subseteq \{1,2\}\times \{a,b\}$. This is a relation, but not a function, because input $1$ has two outputs.

A function as a set of pairs

Let $f:\{1,2,3\}\to \{a,b,c\}$ be given by $f(1)=a$, $f(2)=b$, $f(3)=b$. As a set of ordered pairs, $f=\{(1,a),(2,b),(3,b)\}.$ It is a function but not injective, because two inputs map to $b$, and not surjective onto $\{a,b,c\}$, because $c$ is missed.

Physics-style set

A particle moving on a line can be modelled as a function $x:I\to \mathbb{R}$, where $I$ is a time Interval. The possible positions form a set; the path chooses one position for each time.

Example: solution set

Solving an equation often means finding a set. For

$$x^2=4,$$

the solution set over the real numbers is

$$\{-2,2\}.$$

Over the natural numbers, it would usually be only

$$\{2\}.$$

So the ambient set matters. This is why Number sets and Domain are not pedantic extras; they change answers.