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An arithmetic sequence is a series of numbers where each term after the first is obtained by adding a constant difference, called the common difference, to the preceding term. In our exercise, both sets \( A \) and \( B \) are examples of arithmetic sequences.

• The arithmetic sequence for set \( A \) begins at 2, progresses with a common difference of 2, and ends at 50. The terms are: 2, 4, 6, ..., 50.
• Set \( B \) follows a similar pattern where it starts at 7, with a common difference of 7, and concludes at 49. The terms are: 7, 14, 21, ..., 49.

To find the number of terms in an arithmetic sequence, we use the formula: \[ n = \frac{l - a_1}{d} + 1 \] where \( l \) is the last term, \( a_1 \) is the first term, and \( d \) is the common difference.

For instance, in set \( A \), \( l = 50, a_1 = 2, \) and \( d = 2 \), so there are 25 elements. Similarly, for set \( B \), \( l = 49, a_1 = 7, \) and \( d = 7 \), resulting in 7 elements.

The inclusion-exclusion principle is a fundamental rule used in set theory to accurately count the number of elements in the union of two or more sets. The principle helps avoid double-counting the elements that are present in multiple sets.

This principle can be mathematically expressed as: \[ |A \cup B| = |A| + |B| - |A \cap B| \]Here's a breakdown:

- \(|A|\) is the number of elements in set \( A \).- \(|B|\) is the number of elements in set \( B \).- \(|A \cap B|\) is the number of elements common to both sets, known as the intersection.

In the exercise, when we calculated \(|A \cup B|\), we found: \[ |A \cup B| = 25 + 7 - 3 = 29 \]
This result indicates 29 elements in the smallest subset covering both sets.

Natural numbers are a set of positive integers starting from 1 that do not include fractions or decimals. In most mathematical contexts, they represent the counting numbers starting from 1 (i.e., 1, 2, 3, ...).

In our exercise, the set \( X \) consists of natural numbers from 1 to 50: \[ X = \{n \in \mathbb{N} : 1 \leq n \leq 50\} \]
This set serves as a baseline from which subsets \( A \) and \( B \) are derived based on their specific characteristics as multiples.

Understanding this definition and its application is crucial because operations on subsets \( A \) and \( B \) depend on their nature as members of the set of natural numbers.

In mathematics, a multiple of a number is a product of that number and an integer. Multiples extend indefinitely as you continue to multiply by 1, 2, 3, and so on. In simpler terms, multiples of a number are its repeated addition results.

For example:

• Multiples of 2 in set \( X \) are numbers like 2, 4, 6, up to 50.
• Multiples of 7 in \( X \) are numbers such as 7, 14, 21, up to 49.

Common multiples of two numbers, for instance, 2 and 7, are known as subsequent multiples (in this case, 14, 28, 42) which appear in both sequences derived from multiplying by their base integer together.

Understanding how to identify and list multiples is essential when evaluating intersections and unions of sets comprising multiples.