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Set-Builder Notation

- {x|-3 ≤ x ≤ 16, x ∈ ℤ}

The set of numbers such that…

x is greater than or equal to -3 and less than or equal to 16

And x is an element of the set of integers

Set-Builder Notation

- Describe the set of numbers using set-builder notation. {8, 9, 10, 11, 12, …}
- The set of numbers is always considered x unless otherwise stated
- So we start with: {x|
- This means “the set of number x such that…”

Set-Builder Notation At this point we have: {x| x ≥ 8

- Describe the set of numbers using set-builder notation. {8, 9, 10, 11, 12, …}
- This set only has numbers starting at 8 and increasing
- We write that as an inequality: x ≥ 8
- This includes all the numbers in the set!

Set-Builder Notation

- Describe the set of numbers using set-builder 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

Set-Builder Notation

- Describe the set of numbers using set-builder 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

Set-Builder Notation

- 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}

Set-Builder Notation

- Example 3: All multiples of 3
- In this case, x is equal to 3 times any number
- We write this as x = 3n

- In this case, multiples of 3 can only be an integer (positive or negative whole numbers or zero)
- {x| x = 3n, x ∈ ℤ}

Practice

- {1, 2, 3, 4, 5, …}
- x ≤ 3
- -4 < x ≤ 14
- All multiple of ∏

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