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Sets - Power Set-advanced

Grade 9CBSE

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

🔑Concepts

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The Power Set of a set AA, denoted by P(A)P(A), is defined as the collection of all subsets of AA. This includes the empty set ∅\emptyset and the set AA itself.

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If a set AA has nn elements, then the number of elements in the power set P(A)P(A) is given by 2n2^n.

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An element xx belongs to the power set P(A)P(A) if and only if xx is a subset of AA. Mathematically: X∈P(A)  ⟺  X⊆AX \in P(A) \iff X \subseteq A.

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The empty set ∅\emptyset is always an element of any power set P(A)P(A), and it is also a subset of any power set P(A)P(A).

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The number of non-empty subsets of a set AA is 2n−12^n - 1. Similarly, the number of proper subsets of AA is 2n−12^n - 1 (excluding the set itself).

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For the empty set ∅\emptyset, the power set is P(∅)={∅}P(\emptyset) = \{\emptyset\}. Note that n(P(∅))=20=1n(P(\emptyset)) = 2^0 = 1.

📐Formulae

n(P(A))=2n(A)n(P(A)) = 2^{n(A)}

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

Number of non-empty subsets=2n−1\text{Number of non-empty subsets} = 2^n - 1

P(A)={X:X⊆A}P(A) = \{X : X \subseteq A\}

💡Examples

Problem 1:

Given the set A={1,2,3}A = \{1, 2, 3\}, list all the elements of P(A)P(A) and verify the total number of elements.

Solution:

The subsets of A={1,2,3}A = \{1, 2, 3\} are:

  1. The empty set: ∅\emptyset
  2. Single-element subsets: {1},{2},{3}\{1\}, \{2\}, \{3\}
  3. Two-element subsets: {1,2},{2,3},{1,3}\{1, 2\}, \{2, 3\}, \{1, 3\}
  4. The set itself: {1,2,3}\{1, 2, 3\}

Therefore, P(A)={∅,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}}P(A) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{2, 3\}, \{1, 3\}, \{1, 2, 3\}\}.

The number of elements in AA is n(A)=3n(A) = 3. The number of elements in P(A)P(A) is 23=82^3 = 8. Counting the listed subsets, we have 1+3+3+1=81 + 3 + 3 + 1 = 8 elements.

Explanation:

To find the power set, we systematically list all subsets of size 0 to nn. The formula 2n2^n confirms we have found all possible subsets.

Problem 2:

If A=∅A = \emptyset, find n(P(P(P(A))))n(P(P(P(A)))).

Solution:

Step 1: Find n(A)n(A). Since A=∅A = \emptyset, n(A)=0n(A) = 0. Step 2: Find n(P(A))n(P(A)). n(P(A))=2n(A)=20=1n(P(A)) = 2^{n(A)} = 2^0 = 1 Step 3: Find n(P(P(A)))n(P(P(A))). n(P(P(A)))=2n(P(A))=21=2n(P(P(A))) = 2^{n(P(A))} = 2^1 = 2 Step 4: Find n(P(P(P(A))))n(P(P(P(A)))). n(P(P(P(A))))=2n(P(P(A)))=22=4n(P(P(P(A)))) = 2^{n(P(P(A)))} = 2^2 = 4

Explanation:

This problem uses the power set cardinality formula iteratively. Each 'P' indicates a new power set operation, doubling the exponent based on the previous result.

Problem 3:

A set SS has 63 proper subsets. Find the number of elements in SS.

Solution:

Let the number of elements in set SS be nn. The formula for the number of proper subsets is 2n−12^n - 1. According to the problem: 2n−1=632^n - 1 = 63 2n=63+12^n = 63 + 1 2n=642^n = 64 We know that 64=2664 = 2^6. Therefore: 2n=262^n = 2^6 n=6n = 6 The set SS has 6 elements.

Explanation:

A proper subset is any subset except the set itself. By setting the formula 2n−12^n - 1 equal to the given value, we can solve for nn using powers of 2.