Set - Cardinality of a set

In mathematics, a set can be thought of as any collection of distinct things considered as a whole. Though a simple idea, it is nevertheless one of the most important and fundamental concepts in modern mathematics, and the study of the structure of possible sets, set theory, is quite rich. Set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as primary school. It is the language in which modern mathematics is described. Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from wh ...

Including:

• Set - Definition
• Set - Describing sets
• Set - Descriptions using words or lists
• Set - Descriptions using mathematical notation
• Set - Set membership
• Set - Cardinality of a set
• Set - Subsets
• Set - Special sets
• Set - Unions
• Set - Intersections
• Set - Complements

Read more here: » Set: Encyclopedia - Set

Each of the sets described above has a definite number of members; for example, the set A has four members, while the set B has three members. A set can also have zero members. Such a set is called the empty set (or the null set) and is denoted by the symbol ø. For example, the set A of all three-sided squares has zero members, and thus A = ø. Like the number zero, though seemingly trivial, the empty set turns out to be quite important in mathematics. For more information on the empty set see Empty set. A set can also have an infinite number of members; for exam ...

See also:

Set, Set - Definition, Set - Describing sets, Set - Descriptions using words or lists, Set - Descriptions using mathematical notation, Set - Set membership, Set - Cardinality of a set, Set - Subsets, Set - Special sets, Set - Unions, Set - Intersections, Set - Complements

Read more here: » Set: Encyclopedia II - Set - Cardinality of a set

Set - Descriptions using words or lists. Not all sets have precise descriptions of any sort; they may simply be arbitrary collections, with no expressible "rule" saying what elements are in or out. Some sets may be described in words, for example: A is the set whose members are the first four positive whole numbers. B is the set whose members ar ...

See also:

Set, Set - Definition, Set - Describing sets, Set - Descriptions using words or lists, Set - Descriptions using mathematical notation, Set - Set membership, Set - Cardinality of a set, Set - Subsets, Set - Special sets, Set - Unions, Set - Intersections, Set - Complements

Read more here: » Set: Encyclopedia II - Set - Describing sets

Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B − A, (or B \ A) is the set of all elements which are members of B, but not members of A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing green fr ...

See also:

Set, Set - Definition, Set - Describing sets, Set - Descriptions using words or lists, Set - Descriptions using mathematical notation, Set - Set membership, Set - Cardinality of a set, Set - Subsets, Set - Special sets, Set - Unions, Set - Intersections, Set - Complements

Read more here: » Set: Encyclopedia II - Set - Complements

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B  =  ø, then A and B are said to be disjoint. Examples: {1, 2} ∩ {red, white} = ø {1, 2, green} ∩ {red, white, green} = {green} {1, 2} ∩
...

See also:

Set, Set - Definition, Set - Describing sets, Set - Descriptions using words or lists, Set - Descriptions using mathematical notation, Set - Set membership, Set - Cardinality of a set, Set - Subsets, Set - Special sets, Set - Unions, Set - Intersections, Set - Complements

Read more here: » Set: Encyclopedia II - Set - Intersections

If every member of the set A is also a member of the set B, then A is said to be a subset of B, written , also pronounced A is contained in B. Equivalently, we can write , read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by is called inclusion or containment. If A is a subset of but not equal to B, then A is called a proper subset of B, written (A is a proper subset ...

See also:

Set, Set - Definition, Set - Describing sets, Set - Descriptions using words or lists, Set - Descriptions using mathematical notation, Set - Set membership, Set - Cardinality of a set, Set - Subsets, Set - Special sets, Set - Unions, Set - Intersections, Set - Complements

Read more here: » Set: Encyclopedia II - Set - Subsets

A set is a collection of objects considered as a whole. The objects of a set are called elements or members. The elements of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters, A, B, C, etc. Two sets A and B are said to be equal, written A = B, if they have the same members. As opposed to a multiset and a real-life collection, a set cannot cont ...

See also:

Set, Set - Definition, Set - Describing sets, Set - Descriptions using words or lists, Set - Descriptions using mathematical notation, Set - Set membership, Set - Cardinality of a set, Set - Subsets, Set - Special sets, Set - Unions, Set - Intersections, Set - Complements

Read more here: » Set: Encyclopedia II - Set - Definition

There are some sets which hold great mathematical importance and are referred to with such regularity that they have acquired special names to identify them. One of these is the empty set. Some special sets of numbers include: denotes the set of all natural numbers. That is to say, = {1, 2, 3, ...}, or sometimes = {0, 1, 2, 3, ...}. denotes the set of all integers (whether positive, negative or zero). So = {..., -2, -1, 0, 1, 2, ...}. denotes the set of all rational numbers (that is, the set of all proper ...

See also:

Set, Set - Definition, Set - Describing sets, Set - Descriptions using words or lists, Set - Descriptions using mathematical notation, Set - Set membership, Set - Cardinality of a set, Set - Subsets, Set - Special sets, Set - Unions, Set - Intersections, Set - Complements

Read more here: » Set: Encyclopedia II - Set - Special sets

There are several ways to construct new sets from existing ones. Two sets can be "added" together. The union of A and B, denoted by A U B, is the set of all things which are members of either A or B. Examples: {1, 2} U {red, white} = {1, 2, red, white} {1, 2, green} U {red, white, green} = {1, 2, red, white, green} {1, 2} U {1, ...

See also:

Set, Set - Definition, Set - Describing sets, Set - Descriptions using words or lists, Set - Descriptions using mathematical notation, Set - Set membership, Set - Cardinality of a set, Set - Subsets, Set - Special sets, Set - Unions, Set - Intersections, Set - Complements

Read more here: » Set: Encyclopedia II - Set - Unions