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Sets - Intervals as subsets of R

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Intervals are specific subsets of the set of real numbers R\mathbb{R} that represent a continuous range of values between two endpoints.

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An Open Interval (a,b)(a, b) includes all real numbers xx such that a<x<ba < x < b. The endpoints aa and bb are not included.

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A Closed Interval [a,b][a, b] includes all real numbers xx such that a≤x≤ba \le x \le b. The endpoints aa and bb are included.

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Semi-open or Semi-closed intervals include only one of the endpoints. For example, [a,b)[a, b) includes aa but not bb, represented as a≤x<ba \le x < b.

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The notation (−∞,a](-\infty, a] represents all real numbers less than or equal to aa, and (a,∞)(a, \infty) represents all real numbers greater than aa.

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The Length of any interval (a,b)(a, b), [a,b][a, b], [a,b)[a, b), or (a,b](a, b] is defined by the difference between the endpoints.

📐Formulae

(a,b)={x∈R:a<x<b}(a, b) = \{x \in \mathbb{R} : a < x < b\}

[a,b]={x∈R:a≤x≤b}[a, b] = \{x \in \mathbb{R} : a \le x \le b\}

[a,b)={x∈R:a≤x<b}[a, b) = \{x \in \mathbb{R} : a \le x < b\}

(a,b]={x∈R:a<x≤b}(a, b] = \{x \in \mathbb{R} : a < x \le b\}

Length of interval =b−a\text{Length of interval } = b - a

💡Examples

Problem 1:

Write the set {x:x∈R,−4<x≤6}\{x : x \in \mathbb{R}, -4 < x \le 6\} as an interval and find its length.

Solution:

The interval is (−4,6](-4, 6]. The length is 6−(−4)=106 - (-4) = 10.

Explanation:

Since xx is strictly greater than −4-4, we use a parenthesis (( at −4-4. Since xx is less than or equal to 66, we use a square bracket ]] at 66. The length is the difference between the upper and lower limits: b−ab - a.

Problem 2:

Express the interval [−3,5)[-3, 5) in set-builder form.

Solution:

{x:x∈R,−3≤x<5}\{x : x \in \mathbb{R}, -3 \le x < 5\}

Explanation:

The square bracket at −3-3 indicates that −3-3 is included (≤\le), while the parenthesis at 55 indicates that 55 is excluded (<<).

Problem 3:

Represent the set of all real numbers greater than 77 using interval notation.

Solution:

(7,∞)(7, \infty)

Explanation:

Numbers greater than 77 start from 77 (exclusive) and continue indefinitely towards positive infinity. Infinity is always paired with a parenthesis since it is not a specific reachable number.

Intervals as subsets of R Class 11 Notes & Examples