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 $\varnothing$. $$\bar{X} \cup \bar{Y}$$

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Key Concepts

Set Union

The union of two sets is the set containing all elements that belong to at least one of the sets. It is used to combine elements from different sets, ensuring that each element is included only once regardless of how many times it might appear.

Set-Builder Notation

Set-builder notation is a concise way to define a set by specifying a property that its members must satisfy. It is particularly useful for describing infinite sets or sets with a clear defining property without listing all elements individually.

Universe

In set theory, the universe refers to the overall set of elements under consideration from which subsets are taken. It serves as the domain against which complements of sets are defined, meaning that the complement of a set includes all elements in the universe that are not in that set.

Set Complement

The complement of a set is defined as all the elements in the universe that are not contained in the set. It is a fundamental concept that allows for the expression of exclusion and is critical when combining set operations such as union and intersection.

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