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→Examples: add a concrete example of operations on sets with an explicit list of elements
== Examples ==
== Examples ==
Let <math>S</math> be the set <math>\{a, b, d, f, h\}</math>, and let <math>T</math> be the set <math>\{d, e, f\}</math>
*The cardinality of <math>S</math> is <math>5</math>.
*The cardinality of <math>T</math> is <math>3</math>.
*The intersection, <math>S \cap T</math> is the set <math>\{d, f\}</math>. This is because <math>d</math> and <math>f</math> are the only two elements in both <math>S</math> and <math>T</math>.
*The union, <math>S \cup T</math> is the set <math>\{a, b, d, e, f, h\}</math>. These six elements are in the union because they belong to <math>S</math>, to <math>T</math>, or to both <math>S</math> and <math>T</math>.
*The set-theoretic difference <math>S \setminus T</math> (alternative notation: <math>S - T</math>) is the set <math>\{a, b, h\}</math> containing all the elements in <math>S</math> that are '''not''' in <math>T</math>.
*The power set of <math>T</math>, <math>\mathcal{P}(T)</math>, consists of all subsets of <math>T</math> including the empty set and <math>T</math> itself. One can check that <math>\mathcal{P}(T) = \{\emptyset, \{d\}, \{e\}, \{f\}, \{d, e\}, \{d, f\}, \{e, f\}, \{d, e, f\}\}</math>. ''Exercise: How many elements are in <math>\mathcal{P}(S)</math>?''
Let <math>A</math> be the set of all [[prime number]]s, and let <math>B</math> be the set of all [[even number]]s. What is the intersection, <math>A \cap B</math>? It is a set containing one element, 2. Since any even number greater than 2 has two as a proper factor, no even number aside from 2 belongs to <math>A</math>. However, 2 is a prime number. Thus, the intersection is <math>\{2\}</math>.
Let <math>A</math> be the set of all [[prime number]]s, and let <math>B</math> be the set of all [[even number]]s. What is the intersection, <math>A \cap B</math>? It is a set containing one element, 2. Since any even number greater than 2 has two as a proper factor, no even number aside from 2 belongs to <math>A</math>. However, 2 is a prime number. Thus, the intersection is <math>\{2\}</math>.