Let X = {n ∈ N : 1 ≤ n ≤ 50}. If A = {n ∈ X : n is a multiple of 2} and B = {n ∈ X : n is a multiple

Q: Wait, why do we subtract the intersection? Can't we just add $n(A)$ and $n(B)$?

That is a classic trap! If you simply add $n(A)$ and $n(B)$, you are counting the elements that are in both sets twice—once when you count $A$, and again when you count $B$. To get the correct total for the union, we must subtract the intersection $n(A \cap B)$ exactly once to correct this double-counting.

Q: How do I know if I should use the Least Common Multiple (LCM) or the Greatest Common Divisor (GCD) for the intersection?

Great question! The intersection $A \cap B$ consists of numbers that are multiples of $2$ AND multiples of $7$. For a number to be a multiple of both, it must be a multiple of their LCM. The GCD would give you the largest number that divides both, which is not what we need here. Always think: 'multiple of $x$ AND multiple of $y$' means 'multiple of $\text{lcm}(x, y)$'.

Q: Why do we use the floor function $\lfloor \frac{50}{n} \rfloor$ to count the multiples?

The floor function is the most efficient way to count how many multiples of a number $k$ exist up to a limit $N$. Since we are dealing with natural numbers, $\frac{N}{k}$ gives the number of times $k$ fits into $N$. The integer part of that division tells us exactly how many multiples there are. For example, $\lfloor \frac{50}{7} \rfloor = 7$ tells us there are $7$ multiples of $7$ in the range $1$ to $50$.

Question

Let $X = \{n \in N : 1 \le n \le 50\}$. If $A = \{n \in X : n 	ext{ is a multiple of } 2\}$ and $B = \{n \in X : n 	ext{ is a multiple of } 7\}$, then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in front of a vast, organized box containing exactly fifty marbles, each labeled with a natural number from $1$ to $50$. This is our universal set, $X$.

We define set $A$ as the collection of all even numbers—the multiples of $2$—and set $B$ as the collection of all multiples of $7$. Our mission is to find the size of the smallest subset of $X$ that contains both of these groups.

### The Power of the Floor Function

To find the number of elements in set $A$, we use the elegance of the floor function. We know that the number of multiples of $k$ in the range $1$ to $N$ is given by $\lfloor \frac{N}{k} floor$.

For set $A$, we calculate:
$$n(A) = \left\lfloor \frac{50}{2} ightfloor = 25$$

Similarly, for set $B$, we calculate:
$$n(B) = \left\lfloor \frac{50}{7} ightfloor = 7$$

### The Trap of Double Counting

The question asks for the smallest subset containing both $A$ and $B$, which is the union of the two sets, $A \cup B$. You might be tempted to simply add $n(A) + n(B)$, which would be $25 + 7 = 32$.

However, if you do that, you have counted the numbers that are multiples of both $2$ and $7$ twice. These are the numbers that live in the overlap, the intersection $A \cap B$.

### The Elegance of Inclusion-Exclusion

To find the intersection $A \cap B$, we look for numbers that are multiples of both $2$ and $7$. A number is a multiple of both if and only if it is a multiple of their least common multiple, $	ext{lcm}(2, 7) = 14$.

Using our floor function again:
$$n(A \cap B) = \left\lfloor \frac{50}{14} ightfloor = 3$$

Now, we apply the **Principle of Inclusion-Exclusion**:
$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

Substituting our values into the equation:
$$n(A \cup B) = 25 + 7 - 3$$

The arithmetic is straightforward:
$$n(A \cup B) = 29$$

We have successfully navigated the trap and found that there are **29** elements in the smallest subset containing both $A$ and $B$.

Animated Solution for Mathematics - Sets and Relations: Let $X = {n \in N : 1 \leq n \leq 50}$ . If $A = {n \in X : n is a multiple of 2}$ and $B = {n \in X : n is a multiple of 7}$ , then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is

Enter Numerical Value:

Visualized Solution

Defining the Universal Set $X$

Identifying Set $A$ : Multiples of $2$

Calculating $n (A)$

Identifying Set $B$ : Multiples of $7$

Calculating $n (B)$

The Logic of the Smallest Subset

Finding the Intersection $A \cap B$

Calculating $n (A \cap B)$

Applying the Inclusion-Exclusion Principle

Raw Substitution

Final Calculation

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The Power of the Floor Function

The Trap of Double Counting

The Elegance of Inclusion-Exclusion

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Enter Numerical Value:

Visualized Solution

Defining the Universal Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X

Identifying Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A: Multiples of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(A)

Identifying Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B: Multiples of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 7

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n(B)