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

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Key Concepts

Universe

The universe in set theory is the well-defined set that contains all objects under consideration for a particular discussion or problem. In the context of this example, it is the set of all positive integers, which forms the domain in which complements of sets are determined.

Set Complement

The complement of a set relative to a given universe is the collection of all elements in the universe that are not in the set. It is usually denoted by a bar or a superscript c (e.g., X?) and is a key concept in understanding how sets interact within a universal context.

Intersection

The intersection of two sets is the set containing all elements that are common to both sets. It is an operation that helps identify shared elements between sets and is symbolized by the symbol ?. In problems like this, intersections are often taken after complement operations to form combined conditions.

Set-builder Notation

Set-builder notation is a concise way of specifying sets by defining a property that its members must satisfy, rather than listing out each member explicitly. It is especially useful for describing infinite sets, such as the set of positive even integers in this problem, and plays a central role in formal mathematical definitions.

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