Set-Builder Notation

1. Set-Builder Notation

2. 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

3. 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…”

4. Set-Builder Notation • 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! • At this point we have: {x| x ≥ 8

5. 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

6. 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

7. 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}

8. 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}

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