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I think the real-world examples can be together with the abstract ones.
== 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>.
Let <math>C</math> be the set of all positive integers that are multiples of <math>3</math>, and let <math>D</math> be the set of all positive integers that have a remainder of <math>1</math> when divided by <math>3</math>. The intersection, <math>C \cap D</math>, is the empty set, because no positive integer that is a multiple of <math>3</math> will have remainder <math>1</math> when divided by <math>3</math>. However, if we take <math>B</math> to be the set of even numbers, as before, then <math>C \cap B</math> is not empty. ''Exercise: What are the elements of <math>C \cap B</math>?''
== Examples of sets in the real world ==
There is the set of unborn children who were [[abortion|aborted]], about which striking conclusions can be drawn. Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements. Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.
There is the set of unborn children who were [[abortion|aborted]], about which striking conclusions can be drawn. Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements. Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.
Another striking example is the how traditional marriage provides a greater set than otherwise: the union of <math>A = \{a, b, c, d\}\,</math> and <math>B = \{a, b, c, e\}\,</math> is merely <math>\{a, b, c, d, e\}\,</math>, while the union of a man, <math>M = \{a, b, c, d\}\,</math> and a woman <math>W = \{e, f, g, h\}\,</math>, is <math>\{a, b, c, d, e, f, g, h\}\,</math>, which is a broader and more diverse set.
Define "A" as the set of all false assertions, and "B" as the set of all the Bible passages. Since there are no [[Counterexamples to the Bible]], the intersection between set A and set B is the [[empty set]]. On the other hand, if "C" is set of evolutionary assertions, then the intersection between set A and set C is not empty because it contains [[Counterexamples to Evolution]].
Define "A" as the set of all false assertions, and "B" as the set of all the Bible passages. Since there are no [[Counterexamples to the Bible]], the intersection between set A and set B is the [[empty set]]. On the other hand, if "C" is set of evolutionary assertions, then the intersection between set A and set C is not empty because it contains [[Counterexamples to Evolution]].