Suppose A1, A2, .... A30 are thirty...

MathematicsTypes of Sets and Set OperationsJEE Advanced 1981Easy

Suppose $A_{1}, A_{2}, \dots, A_{30}$ are thirty sets each with five elements and $B_{1}, B_{2}, \dots, B_{n}$ are $n$ sets each with three elements. Let $⋃_{i = 1}^{30} A_{i} = ⋃_{j = 1}^{n} B_{j} = S$ . Assume that each element of $S$ belongs to exactly ten of the $A_{i}$ 's and to exactly nine of the $B_{j}$ 's. Find $n$ .

Answer:45

Visualized Solution (English)

Understanding the Problem

Given sets: $A_{1}, A_{2}, \dots, A_{30}$ with $n (A_{i}) = 5$ .
Given sets: $B_{1}, B_{2}, \dots, B_{n}$ with $n (B_{j}) = 3$ .
Union: $⋃ A_{i} = ⋃ B_{j} = S$ .
Each element of $S$ is in exactly $10$ of $A_{i}$ and $9$ of $B_{j}$ .

The Counting Principle

Principle: If each element of $S$ belongs to exactly $k$ sets $X_{1}, X_{2}, \dots, X_{m}$ , then:
$n (S) = \frac{\sum _{i = 1}^{m} n ( X _{i} )}{k}$

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n (S) using .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A_{i}

Total elements in $A_{i}$ (with repetition) $= 30 \times 5 = 150$ .
Since each element is in $10$ sets, $n (S) = \frac{150}{10} = 15$ .

Setting up the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B_{j} Equation

Total elements in $B_{j}$ (with repetition) $= n \times 3 = 3 n$ .
Since each element is in $9$ sets, $n (S) = \frac{3 n}{9} = \frac{n}{3}$ .

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Equating the two values of $n (S)$ :
$\frac{n}{3} = 15$
$n = 15 \times 3 = 45$

Key Takeaways

Key Concept: Double counting principle in Set Theory.
Formula: $n (S) = \frac{Total elements with repetition}{Repetition factor}$
Final Answer: $n = 45$

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Visualized Solution (English)

Understanding the Problem

The Counting Principle

Finding $n (S)$ using $A_{i}$

Setting up the $B_{j}$ Equation

Solving for $n$

Key Takeaways

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Visualized Solution (English)

Understanding the Problem

The Counting Principle

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(S) using .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Ai​

Setting up the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Bj​ Equation

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Key Takeaways

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