1.2: Basics of sets and real numbers

To begin studying mathematics at the university level, we really need to know about sets.

Definition of sets¶

Sets are collections of distinct objects. Quite often these objects will be numbers. For example,

is the set of all even numbers between 2 and 8. The numbers 2, 4, 6, 8 are called elements of the set $A$. Here we’re describing the set by writing down every element inside curly bracelets.

Objects don’t always need to be numbers. We can have a set of colors that the cone cells in our eyes can detect:

The colors red, green, and blue are the elements of set $B$.

Membership of sets¶

To denote that objects belong to sets, we use the symbol $\in$. For example, recall that the number 4 is an element of the set $A$, and that red is an element of $B$. We denote this as

If an object does not belong to a set, we write down the crossed-out version of the symbol above. The number 5 and the color yellow are not elements of $A$ and $B$, so we write

Set-builder notation¶

So far we’ve only looked at denoting sets by writing down all of their elements, which is called the enumerative notation. This is often not very practical for our purposes, which is why we’ll now see a different notation called set-builder notation.

Suppose we want to define the set $C$ as the collection of all numbers in $A$ multiplied by 3. We could use the enumerative notation to write down all the elements

The set-builder notation for $C$ is given by

There are two pieces of information here. The “$3x$”, written on the left of the colon, denotes the elements of $C$. The “$x\in A$”, on the right of the colon, denotes the condition on the variable $x$. Hence we can interpret the notation above as saying:

More examples need to be added.

The real numbers $\mathbb{R}$¶

There is one set we will see most frequently in this course: the set of real numbers. This set is denoted by $\mathbb{R}$.

The real numbers are the collection of every number we’ve ever learned in school, which includes:

• integers $-2,-1,0,1,2,\ldots$

• fractions $-\frac{3}{2}, \frac{1}{4},\ldots$

• irrational numbers $\sqrt{2}, -\sqrt{3}, \ldots$.

Let’s practice with the set-builder notation we’ve just learned. Suppose the set $I$ is defined to be the collection of all real numbers that are bigger than zero, i.e. the set of positive numbers. How would it be denoted using set-builder notation? We would write down something like:

How about the set $J$ of real numbers between zero and one? We would write down