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In certain contexts we may consider all of our sets as being subsets of some given universal set. For instance, if we are investigating properties of the real numbers R (and subsets of R), then we may take R as our universal set. It is important to realise that a universal set is only temporarily defined by the context; there is no such thing as a "universal" universal set, "the set of everything" (see Paradoxes below).
Given a universal set U and a subset A of U, we may define the complement of A (in U) as
i.e., the set of all members of U which are not members of A. Thus with A, B and C as in the section on subsets, if B is the universal set, then C' is the set of even integers, while if A is the universal set, then C' is the set of all real numbers that are either even integers or not integers at all.
The collection {A : A ⊆ U} of all subsets of a given universe U is called the power set of U. (See axiom of power set.) It is denoted P(U); the "P" is sometimes in a fancy font.
Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B.
The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B.
Finally, the relative complement of B relative to A, also known as the set theoretic difference of A and B, is the set of all objects that belong to A but not to B. It is written as A \ B. Symbolically, these are respectively
Notice that A doesn't have to be a subset of B for B \ A to make sense; this is the difference between the relative complement and the absolute complement from the previous section.
To illustrate these ideas, let A be the set of left-handed people, and let B be the set of people with blond hair. Then A ∩ B is the set of all left-handed blond-haired people, while A ∪ B is the set of all people who are left-handed or blond-haired or both. A \ B, on the other hand, is the set of all people that are left-handed but not blond-haired, while B \ A is the set of all people that have blond hair but aren't left-handed.
Now let E be the set of all human beings, and let F be the set of all living things over 1000 years old. What is E ∩ F in this case? No human being is over 1000 years old, so E ∩ F must be the empty set {}.
For any set A, the power set is a Boolean algebra under the operations of union and intersection.
Intuitively, an ordered pair is simply a collection of two objects such that one can be distinguished as the first element and the other as the second element, and having the fundamental property that, two ordered pairs are equal if and only if their first elements are equal and their second elements are equal.
Formally, an ordered pair with first coordinate a, and second coordinate b, usually denoted by (a, b), is defined as the set a}, {a, b}}.
It follows that, two ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d.
The notation (a, b) is also used to denote an open interval on the real number line, the context will make it clear which meaning is meant.
If A and B are sets, then the Cartesian product (or simply product) is defined to be:
That is, A × B is the set of all ordered pairs whose first coordinate is an element of A and whose second coordinate is an element of B.
We can extend this definition to a set A × B × C of ordered triples, and more generally to sets of ordered n-tuples for any positive integer n. It is even possible to define infinite Cartesian products, but to do this we need a more recondite definition of the product.
Cartesian products were first developed by René Descartes in the context of analytic geometry. If R denotes the set of all real numbers, then R2 := R × R represents the Euclidean plane and R3 := R × R × R represents three-dimensional Euclidean space.
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