Element (mathematics)

In mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the set (or class).

Set theory and elements

Writing $A=\{1,2,3,4\}$, means that the elements of the set $A$ are the numbers 1, 2, 3 and 4. Groups of elements of $A$, for example $\{1,2\}$, are subsets of $A$.

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

The elements of a set can be anything. For example, $C=\{{\mbox{red, green, blue}}\}$, is the set whose elements are the colors red, green and blue.

Notation

The relation "is an element of", also called set membership, is denoted by ∈, and writing

$x\in A$

means that $x$ is an element of $A$. Equivalently one can say or write "$x$ is a member of $A$", "$x$ belongs to $A$", "$x$ is in $A$", or "$A$ includes $x$", or "$A$ contains $x$". The negation of set membership is denoted by ∉.

Unfortunately, the terms "$A$ includes $x$" and "$A$ contains $x$" are ambiguous, because some authors also use them to mean "$x$ is a subset of $A$". George Boolos strongly urged that "contains" be reserved for membership only and "includes" for the subset relation.

Cardinality of sets

The number of elements in a particular set is a property known as cardinality, informally this is the size of a set. In the above examples the cardinality of the set $A$ is 4, while the cardinality of the sets $B$ and $C$ 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, $\mathbb {N} =\{1,2,3,4\ldots \}$.

Examples

Using the sets defined above as

• 2 ∈ A
• {3,4} ∈ B
• {3,4} is a member of B
• Yellow ∉ C
• The cardinality of $D=\{2,4,6,8,10,12\}$ is finite and equal to 6.
• The cardinality of $P=\{2,3,5,7,11,13...\}$ (the prime numbers) is infinite.

References

• Paul R. Halmos 1960, Naive Set Theory, Springer-Verlag, NY, ISBN 0-387-90092-6. "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).

• Patrick Suppes 1960, 1972, Axiomatic Set Theory, Dover Publications, Inc. NY, ISBN 0-486-61630-4. Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".