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Sets - Subset-advanced

Grade 9CBSE

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

🔑Concepts

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A set AA is said to be a subset of a set BB if every element of AA is also an element of BB, denoted as A⊆BA \subseteq B.

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If A⊆BA \subseteq B and A≠BA \neq B, then AA is called a proper subset of BB, denoted as A⊂BA \subset B. In this case, BB is called the superset of AA.

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The empty set ∅\emptyset (or {}\{\}) is considered a subset of every set.

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Every set AA is a subset of itself (A⊆AA \subseteq A).

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The collection of all subsets of a set AA is called the Power Set of AA, denoted by P(A)P(A).

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Subsets of the set of Real Numbers R\mathbb{R} can be represented as intervals: Open interval (a,b)={x:a<x<b}(a, b) = \{x : a < x < b\} and Closed interval [a,b]={x:a≤x≤b}[a, b] = \{x : a \le x \le b\}.

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The Universal Set UU is the set that contains all objects under consideration, and all other sets are subsets of UU.

📐Formulae

A⊆B  ⟺  (x∈A  ⟹  x∈B)A \subseteq B \iff (x \in A \implies x \in B)

If n(A)=m, then n(P(A))=2m\text{If } n(A) = m, \text{ then } n(P(A)) = 2^m

Number of proper subsets=2m−1\text{Number of proper subsets} = 2^m - 1

[a,b)={x:a≤x<b}[a, b) = \{x : a \le x < b\}

(a,b]={x:a<x≤b}(a, b] = \{x : a < x \le b\}

💡Examples

Problem 1:

Given the set A={1,2,{3,4},5}A = \{1, 2, \{3, 4\}, 5\}, determine if the following statement is true or false: {3,4}⊂A\{3, 4\} \subset A.

Solution:

The statement is False.

Explanation:

In set AA, the element {3,4}\{3, 4\} is a member of the set, so we write {3,4}∈A\{3, 4\} \in A. For {3,4}\{3, 4\} to be a subset ({3,4}⊂A\{3, 4\} \subset A), the elements 33 and 44 would need to be individual members of AA, but they are not. However, {{3,4}}⊂A\{\{3, 4\}\} \subset A would be true.

Problem 2:

Find the power set P(S)P(S) of the set S={a,b,c}S = \{a, b, c\}.

Solution:

P(S)={∅,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}}P(S) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}, \{a, b, c\}\}

Explanation:

The power set includes all possible subsets. Since n(S)=3n(S) = 3, the number of elements in the power set is 23=82^3 = 8.

Problem 3:

Write the set B={x:x∈R,−3≤x<7}B = \{x : x \in \mathbb{R}, -3 \le x < 7\} as an interval and find its length.

Solution:

Interval form: [−3,7)[-3, 7). Length =7−(−3)=10= 7 - (-3) = 10.

Explanation:

The symbol ≤\le indicates a closed boundary (bracket [[), and << indicates an open boundary (parenthesis )). The length of any interval (a,b)(a, b), [a,b][a, b], [a,b)[a, b), or (a,b](a, b] is given by b−ab - a.

Problem 4:

A set XX has 55 elements. How many proper subsets does it have?

Solution:

25−1=32−1=312^5 - 1 = 32 - 1 = 31

Explanation:

The total number of subsets is 2n2^n. Proper subsets exclude the set itself, so we subtract 11 from the total count. Here n=5n=5.