23.02 Special number sets
Recall that a set is a list of objects, each of which is called an element. Let's review two types of notations we can use to refer to a subset of the real numbers.
Set builder notation
Set builder notation follows a common convention.
| Symbols | $\lbrace x\mid${x∣ | $x>5\rbrace$x>5} |
|---|---|---|
| Read | The set of all numbers $x$x such that | $x$x has a certain property |
Many subsets of the real numbers have abbreviations which are commonly used.
| Set | Letter Abbreviation |
|---|---|
| Real numbers | $\mathbb{R}$ℝ |
| Rational numbers | $\mathbf{Q}$Q |
| Irrational numbers | $\mathbf{Q}'$Q′ |
| Integers | $\mathbb{Z}$ℤ |
| Positive integers | $\mathbb{Z}^+$ℤ+ |
| Natural numbers | $\mathbb{N}$ℕ |
| Empty set | $\varnothing$∅ |
Worked example
Describe the set of even numbers using set builder notation.
Think: The set of even numbers are all natural numbers that are multiples of $2$2. If $x$x is an even number, then $x=2n$x=2n, where $n$n is a natural number. The symbol $\in$∈ means 'is an element of'.
Do: Translate to set builder notation.
| $\lbrace x\mid x=2n,n\in N\rbrace${x∣x=2n,n∈N} |
"the set of all x such that $x=2n$x=2n, where $n$n is an element of the natural numbers" |
Practice questions
Question 1
Is the following statement true or false?
$1$1$\in$∈$\left\{9,8,6,5,1\right\}${9,8,6,5,1}
True
AFalse
B
Question 2
Consider the set $A=\left\{2,4,6,8\right\}$A={2,4,6,8}
Which of the following is the correct set builder notation for $A$A?
{$x$x$|$| $2\le x\le8$2≤x≤8}
A{$x$x$|$|$x$x is an even number}
B{$x$x$|$| $x$x is an even number and $x\ge2$x≥2}
C{$x$x$|$| $x$x is an even number and $2\le x\le8$2≤x≤8}
D