Mathematical Set

Explained:

complement

disjoint

A set is an unordered collection of elements. Elements might be people, days of the week, colors, numbers, etc. We may indicate the elements of a set by listing them between brackets. The set of prime numbers less than 10 is expressed {2, 3, 5, 7}. Alternatively, we may use the notation {x: property} to indicate the set of all elements x satisfying the indicated property. In this manner, the same set of prime numbers less than 10 is expressed as {x: x < 10 and x is prime}. You should be familiar with the following sets:

: the empty set is the set that contains no elements;

: the set of natural numbers {1, 2, 3, };

: the set of integers {, –2, –1, 0, 1, 2, };

: the set of real numbers is the set of all numbers that form the number line; and

: the set of complex numbers is the set of all numbers of the form a + bi, where a and b are real and i is the imaginary number .

We denote intervals of real numbers with parentheses or braces, depending upon whether or not end points are included. The interval (2, 3) is the set {x: 2 < x < 3}. The interval [5, 10) is the set {x: 5 x < 10}.

If a is an element of A, we denote this a A. We read this "a is contained in A." Otherwise, we say "a is not contained in A" and indicate that relationship with a A.

A set B is a subset of A if every element of B is an element of A. We indicate this relationship with the notation B A. The empty set is a subset of every set, and every set is a subset of itself. If B A and A B, the sets A and B are equal, A = B. The union of two sets C and D is the set CD that contains all elements that are either elements of C and/or elements of D. The intersection of two sets C and D is the set CD that contains all elements that are both elements of C and elements of D. Generalizing to more than two sets, let be a collection of sets. We define their union as the set of elements that belong to at least one of the sets . We define the intersection of the sets as the set of elements that belong to every set . We denote such unions and intersections as, respectively

	[1]
	[2]

Two sets are said to be disjoint if their intersection is . Three or more sets are said to be disjoint if each pair of sets is disjoint.

If A B, the complement of A with respect to B is the set B \ A of all elements of B that are not elements of A. Obviously, B \ B = and B \ = B. In a particular context, if it is understood that all complements are being taken with respect to some specific set B, a complement B \ A may be denoted , which is pronounced "the complement of A" or "A complement."

If A , we call A a real set. Its maximum and minimum are the largest and smallest values contained in A, respectively. We denote these max(A) and min(A). Because the maximum or minimum of a set must be elements of that set, a set may not have a maximum or minimum. The interval has neither a maximum nor a minimum. We can generalize the notion of maximum and minimum with the notions of supremum and infimum. If A is real, its supremum, which we denote sup(A), is the smallest real number such that all elements of A are less than or equal to that number. Its infimum, which we denote inf(A), is the largest real number such that all elements of A are greater than or equal to that number. The supremum or infimum of a set need not be elements of the set. For example, the set defined by the interval [0,1) has infimum 0 and supremum 1. The interval has no infimum in , but it has supremum 100.