Use set-builder notation to describe each set. See Example 2. (More than one description is possible.) {2, 4, 6, 8}

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Identify the pattern in the given set {2, 4, 6, 8}. Notice that all elements are even numbers and they increase by 2 each time.

Express the elements as multiples of 2. For example, 2 = 2 \(\times\) 1, 4 = 2 \(\times\) 2, 6 = 2 \(\times\) 3, and 8 = 2 \(\times\) 4.

Write the set in set-builder notation by defining a variable, say x, and stating the condition it must satisfy. For instance, x is an even number between 2 and 8 inclusive.

Formulate the set-builder notation as \(\{ x \mid x = 2n, n \in \mathbb{Z}, 1 \leq n \leq 4 \}\), where \(n\) is an integer that generates the elements of the set.

Alternatively, you can describe the set as \(\{ x \mid x \text{ is even and } 2 \leq x \leq 8 \}\), which also captures the same elements.

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주요 개념

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Set-Builder Notation

Set-builder notation is a concise way to describe a set by specifying a property that its members satisfy. Instead of listing elements, it defines the set as {x | condition on x}, meaning 'the set of all x such that the condition holds.' This notation is useful for describing infinite or large sets.

Properties of Even Numbers

Even numbers are integers divisible by 2 without a remainder. Recognizing that the set {2, 4, 6, 8} consists of even numbers helps in expressing the set using a condition like 'x is an even number between 2 and 8.' Understanding this property aids in forming accurate set-builder descriptions.

Domain Restrictions in Set Definitions

When using set-builder notation, it's important to specify the domain or range of elements considered, such as integers within a certain interval. For example, restricting x to integers between 2 and 8 ensures the set includes only the listed elements, avoiding unintended members.