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Set-Builder Notation. 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.

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Set builder notation1
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 notation2
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 notation3
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


  • Set builder notation4
    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 notation5
    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 notation6
    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 notation7
    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
    Practice

    • {1, 2, 3, 4, 5, …}

    • x ≤ 3

    • -4 < x ≤ 14

    • All multiple of ∏


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