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Question:

Write the following intervals in set-builder form :

(i) (-3, 0) (ii) [6 , 12] (iii) (6, 12] (iv) [-23, 5)

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Last updated date: 14th Jun 2024
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Answer
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Solution:

Hint:

Set-builder notation is a shorthand way to describe a set by indicating the properties that its members must satisfy. In the context of intervals:


  • (a, b) represents all numbers between a and b but not including a or b.

  • [a, b] includes the endpoints, meaning it represents all numbers between a and b including a and b themselves.

  • (a, b] or [a, b) includes only one endpoint.


Step by Step Solution:

(i) (–3, 0)

This interval includes all numbers between -3 and 0, but not -3 and 0 themselves.

In set-builder notation:
$ x : -3 < x < 0 $


(ii) [6, 12]

This interval includes all numbers between 6 and 12, and also the numbers 6 and 12.

In set-builder notation:

$ x : 6 \leq x \leq 12 $


(iii) (6, 12]

This interval includes all numbers between 6 and 12, and also the number 12, but not 6.

In set-builder notation:

$ x : 6 < x \leq 12 $


(iv) [–23, 5]

This interval includes all numbers between -23 and 5, and also the numbers -23 and 5.

In set-builder notation:

$ x : -23 \leq x \leq 5 $


Note: Set-builder notation is particularly useful when describing sets that cannot be neatly described as intervals or in roster form. For intervals, it provides a clear understanding of the inclusivity or exclusivity of the endpoints.