Question

Let \(A=\\{2,4, \ldots, 2 n, \ldots)\) and \(\mathrm{B}=\\{3,6, \ldots, 3 n, \ldots)\). Find \(A \cap \mathrm{B}\) and \(\mathrm{A}-\mathrm{B}\).

Short answer

\( A \cap B = \{6, 12, 18, \ldots\} \); \( A - B = \{2, 4, 8, 10, 14, 16, \ldots\} \).

Step-by-step solution

Understand the sets

Set \( A \) is defined as \( \{2,4, \, \ldots, \, 2n, \, \ldots\} \), meaning it contains all even natural numbers. Similarly, set \( B \) is defined as \( \{3,6,\ldots, \, 3n, \, \ldots\} \), containing all multiples of 3.

Determine the intersection (A ∩ B)

To find \( A \cap B \), identify elements that are both multiples of 2 and multiples of 3. The common multiples are the multiples of the least common multiple (LCM) of 2 and 3, which is 6. Thus, \( A \cap B = \{6, 12, 18, \ldots\} \).

Determine the difference (A - B)

To find \( A - B \), list all elements in \( A \) that are not in \( B \). Since \( B \) only contains multiples of 3, exclude from \( A \) those elements that are multiples of 6 (since they belong to both sets). Thus, \( A - B \) contains even numbers that are not multiples of 3, i.e., for integers \( k \), if \( k \) is not divisible by 3, then \( 2k \) is in \( A - B \). This results in \( A - B = \{2, 4, 8, 10, 14, 16, \ldots\} \).

Key concepts

Intersection of Sets

When dealing with sets, one important concept is the *intersection* of sets. Intersection refers to all the elements that are common to both sets. In our exercise, we have two sets:

• Set \( A = \{2, 4, 6, 8, \ldots\} \), which includes all even natural numbers.
• Set \( B = \{3, 6, 9, 12, \ldots\} \), which includes all multiples of 3.

To determine the intersection \( A \cap B \), we look for numbers that appear in both sets. Let’s slow down and think about this. A number that's in both sets must be *even* because it's from set \( A \) and must also be a *multiple of 3* because it's part of set \( B \). Such numbers are multiples of both 2 and 3.

Finding the smallest number that satisfies both conditions is key. The smallest number divisible by both 2 and 3 is the Least Common Multiple (LCM) of these numbers. Let's find that LCM in our next section.

Set Difference

The concept of the set *difference* involves all the elements that belong to one set but not the other. For our exercise, this means finding \( A - B \), which includes elements in \( A \) that are not in \( B \).

• Think about set \( A = \{2, 4, 6, 8, 10, \ldots\} \), which contains all even numbers.
• Set \( B = \{3, 6, 9, 12, \ldots\} \) only contains multiples of 3.

To find \( A - B \) effectively, we need to exclude numbers from \( A \) that are common with \( B \). These common elements are multiples of 6, as we previously identified them as the intersection \( A \cap B \). Consequently, \( A - B \) involves taking set \( A \) and removing numbers like 6, 12, 18, etc.

Thus, \( A - B \) is formed by even numbers that are not divisible by 3, such as 2, 4, 8, 10, 14, and so on. To identify such numbers:

• Pick an even number \( 2k \).
• Verify if \( k \) is a multiple of 3. If not, then \( 2k \) belongs to \( A - B \).

Least Common Multiple

The *Least Common Multiple* (LCM) of two numbers is the smallest number that is a multiple of both numbers. In the context of our exercise, we need to calculate the LCM of 2 and 3 to understand the intersection \( A \cap B \).

For numbers 2 and 3:

• 2 is the simplest even number.
• 3 is the integer that only divides itself or 1.

The *LCM* is calculated by finding the smallest number that is divisible by both. In this case, it's straightforward to list multiples of 2 and 3 and spot the lowest common one:

• For 2: \( 2, 4, 6, 8, 10, \ldots \)
• For 3: \( 3, 6, 9, 12, \ldots \)

The first common value in both lists is 6. Therefore, the LCM is 6.

That's why \( A \cap B \) is made up of \( \{6, 12, 18, \ldots \} \), all multiples of 6. These are the numbers common in both sets as explained in the section about the intersection of sets.