Element (mathematics)

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Element (mathematics)

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an element or member of a set is any one of the distinct objects that make up that set.

ing , means that the elements of the set are the numbers 1, 2, 3 and 4. Groups of elements of , for example , are subset

Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
s of .

Elements can themselves be sets. For example consider the set . The elements of are not 1, 2, 3, and 4. Rather, there are only three elements of , namely the numbers 1 and 2, and the set .

The elements of a set can be anything.

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In mathematics

Mathematics

Set theory and elements

Writing , means that the elements of the set are the numbers 1, 2, 3 and 4. Groups of elements of , for example , are subset

Subset

Notation

The relation

Relation (mathematics)

In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
"is an element of", also called set membership, is denoted by ?, and writing

means that is an element of . Equivalently one can say or write " is a member of ", " belongs to ", " is in ", " lies in ", " includes ", or " contains ". The negation

Negation

In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true....
of set membership is denoted by ?.

Unfortunately, the terms " includes " and " contains " are ambiguous, because some authors also use them to mean " is a subset

Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
of ". Logician George Boolos

George Boolos

George Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology....
strongly urged that "contains" be used for membership only and "includes" for the subset relation only.

Cardinality of sets

The number of elements in a particular set is a property known as cardinality

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of Set ....
, informally this is the size of a set. In the above examples the cardinality of the set is 4, while the cardinality of the sets and is 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of natural numbers, .

Examples

Using the sets defined above as

2 ? A
? B
is a member of B
Yellow ? C
The cardinality of is finite and equal to 6.
The cardinality of (the prime numbers) is infinite.