### Set-Builder Notation

```
{x|-3 ≤ x ≤ 16, x ∈ ℤ}
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Describe the set of numbers using setbuilder notation. {8, 9, 10, 11, 12, …}
The set of numbers is always considered x
unless otherwise stated
◦ This means “the set of number x such that…”
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Describe the set of numbers using setbuilder notation. {8, 9, 10, 11, 12, …}
This set only has numbers starting at 8 and
increasing
We write that as an inequality: x ≥ 8
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This includes all the numbers in the set!
At this point we have: {x| x ≥ 8
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Describe the set of numbers using setbuilder notation. {8, 9, 10, 11, 12, …}
We now have to state what set of number x is
an element of
Since these numbers are positive whole
numbers, the set is W
◦ We can write this as x ∈ W
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Describe the set of numbers using setbuilder notation. {8, 9, 10, 11, 12, …}
We can then put everything together for the
final answer: {x| x ≥ 8, x ∈ W}
Verbally this reads: The set of all x such that
x is greater than or equal to 8 and x is an
element of the set of whole numbers
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Example 2: Write the following in set-builder
notation: x < 7
There’s no stipulation on the numbers as
long as they’re less than 7, so it can be all
real numbers
Therefore: {x| x < 7, x ∈ ℝ}
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Example 3: All multiples of 3
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In this case, x is equal to 3 times any number
◦ We write this as x = 3n
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In this case, multiples of 3 can only be an
integer (positive or negative whole numbers
or zero)
{x| x = 3n, x ∈ ℤ}
1.
{1, 2, 3, 4, 5, …}
2.
x≤3
3.
-4 < x ≤ 14
4.
All multiple of ∏
```