Let the universe be the set Z^+ . Let X= {1,2,3,4,5} and let...

Let the universe be the set $Z^{+} .$ Let $X=$ \{1,2,3,4,5\} and let $Y$ be the set of positive, even integers. In set builder notation, $Y=\left\{2 n \mid n \in Z^{+}\right\} .$ In Exercises $18-27,$ give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as $\varnothing$. Describe $\bar{Y}$ in words.

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Key Concepts

Universal Set

The universal set refers to the complete collection of elements under consideration for a particular discussion, problem, or context. Here, the universal set is the set of all positive integers, meaning that any operations such as forming complements are performed with respect to this set.

Set Complement

The complement of a set consists of all elements in the universal set that are not in the given set. In the context of this problem, the complement of the set of positive, even integers includes all the positive integers that are not even.

Even and Odd Integers

Even integers are integers that can be exactly divided by 2, while odd integers are those that are not divisible by 2. Since the set under consideration comprises positive integers and Y is defined as the set of even integers, the complement of Y is the set of positive odd integers.

Set Builder Notation

Set builder notation is a method of describing a set by specifying the properties that its elements must satisfy, rather than listing them explicitly. This notation is particularly useful for defining infinite sets, such as the set of positive even integers.

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