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 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 | $\varnothing$∅ |
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.
$\lbrace x\mid x=2n,n\in\mathbb{N}\rbrace${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" |
Let set $A=\lbrace x|\quad x\text{ is a triangle number }\rbrace$A={x| x is a triangle number }.
a) Write down the first ten elements in set $A$A.
Think: Triangle numbers are numbers that can be arranged to form a triangle. Some examples are in the table below:
Diagram | Triangle number | Amount being added from previous number |
---|---|---|
$1$1 | ||
$3$3 | $2$2 | |
$6$6 | $3$3 | |
$10$10 | $4$4 |
We can see in the table above that the number being added increases by one each time:
Do: So the first ten triangle numbers are:
Triangle number | $1$1 | $3$3 | $6$6 | $10$10 | $15$15 | $21$21 | $28$28 | $36$36 | $45$45 | $55$55 |
---|---|---|---|---|---|---|---|---|---|---|
Add to get next number | $+2$+2 | $+3$+3 | $+4$+4 | $+5$+5 | $+6$+6 | $+7$+7 | $+8$+8 | $+9$+9 | $+10$+10 |
b) Is $A\subset\mathbb{Z}^+$A⊂ℤ+?
Do: Yes, all the values in set $A$A are positive integers.
Is the following statement true or false?
$1$1$\in$∈$\left\{9,8,6,5,1\right\}${9,8,6,5,1}
True
False
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$2≤x≤8}
{$x$x$|$|$x$x is an even number}
{$x$x$|$| $x$x is an even number and $x\ge2$x≥2}
{$x$x$|$| $x$x is an even number and $2\le x\le8$2≤x≤8}