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Sets - Representation of a Set-advanced

Grade 9CBSE

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

🔑Concepts

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A set is a well-defined collection of distinct objects. Objects in a set are called elements or members. If xx is an element of set AA, we write x∈Ax \in A.

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Roster (Tabular) Form: Elements are listed within braces {}\{ \} and separated by commas. The order of elements is immaterial, and elements are usually not repeated. For example, the set of letters in the word 'MATHEMATICS' is {M,A,T,H,E,I,C,S}\{M, A, T, H, E, I, C, S\}.

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Set-Builder Form: A set is described by a property P(x)P(x) that all its elements satisfy. It is written as {x:P(x)}\{x : P(x)\} or {x∣P(x)}\{x | P(x)\}, which reads as 'the set of all xx such that P(x)P(x) is true'.

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Cardinality: The number of distinct elements in a finite set AA is its cardinal number, denoted by n(A)n(A). For example, if A={2,4,6,8}A = \{2, 4, 6, 8\}, then n(A)=4n(A) = 4.

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Special Sets: Symbols used for standard sets include N\mathbb{N} for Natural numbers, W\mathbb{W} for Whole numbers, Z\mathbb{Z} for Integers, Q\mathbb{Q} for Rational numbers, and R\mathbb{R} for Real numbers.

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The Empty Set: A set containing no elements is called an empty set or null set, represented by ∅\emptyset or {}\{ \}. Note that {∅}\{\emptyset\} is NOT an empty set; it is a singleton set containing the empty set.

📐Formulae

x∈A  ⟹  x belongs to set Ax \in A \implies \text{x belongs to set A}

x∉A  ⟹  x does not belong to set Ax \notin A \implies \text{x does not belong to set A}

n(A)=Number of elements in set An(A) = \text{Number of elements in set A}

N={1,2,3,4,… }\mathbb{N} = \{1, 2, 3, 4, \dots\}

Z={…,−2,−1,0,1,2,… }\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}

💡Examples

Problem 1:

Write the set A={x:x∈Z,x2<20}A = \{x : x \in \mathbb{Z}, x^2 < 20\} in roster form.

Solution:

A={−4,−3,−2,−1,0,1,2,3,4}A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}

Explanation:

We need to find integers xx such that x2x^2 is less than 2020. Testing integers: (−4)2=16<20(-4)^2 = 16 < 20, (−5)2=25>20(-5)^2 = 25 > 20. Similarly, 42=16<204^2 = 16 < 20 and 52=25>205^2 = 25 > 20. The integers satisfying the condition are from −4-4 to 44.

Problem 2:

Represent the set B={1,4,9,16,25,… }B = \{1, 4, 9, 16, 25, \dots\} in set-builder form.

Solution:

B={x:x=n2,n∈N}B = \{x : x = n^2, n \in \mathbb{N}\}

Explanation:

The elements 1,4,9,16,251, 4, 9, 16, 25 are the squares of natural numbers: 12,22,32,42,521^2, 2^2, 3^2, 4^2, 5^2. Thus, the general property is x=n2x = n^2 where nn is a natural number.

Problem 3:

Given C={x:x=nn2+1,n∈N,n≤3}C = \{x : x = \frac{n}{n^2+1}, n \in \mathbb{N}, n \le 3\}, find the roster form and the cardinal number n(C)n(C).

Solution:

Roster form: C={12,25,310}C = \{\frac{1}{2}, \frac{2}{5}, \frac{3}{10}\}. Cardinal number: n(C)=3n(C) = 3.

Explanation:

Substitute n=1,2,3n = 1, 2, 3 into the formula nn2+1\frac{n}{n^2+1}: For n=1,x=112+1=12n=1, x = \frac{1}{1^2+1} = \frac{1}{2} For n=2,x=222+1=25n=2, x = \frac{2}{2^2+1} = \frac{2}{5} For n=3,x=332+1=310n=3, x = \frac{3}{3^2+1} = \frac{3}{10} There are 3 distinct elements.