Element of a set

 ( Basic Concepts In Set Theory ) 

In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called containing the first four positive , one could say that "3 is an element of ", expressed notationally as $3\; \backslash in\; A$.

---

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

Sets can themselves be elements. For example, consider the set $B\; =\; \backslash \{1,\; 2,\; \backslash \{3,\; 4\backslash \}\backslash \}$. 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 $\backslash \{3,\; 4\backslash \}$.

The elements of a set can be anything. For example the elements of the set $C\; =\; \backslash \{\backslash mathrm\{\backslash color\{Red\}red\},\; \backslash mathrm\{12\},\; B\backslash \}$ are the color red, the number 12, and the set .

In logic, a set can be defined in terms of the membership of its elements as $(x\; \backslash in\; y)\; \backslash leftrightarrow\; \backslash forall\; xP\_x:\; x\; \backslash in\; \backslash mathfrak\; D\; y$. This basically means that there is a general predication of x called membership that is equivalent to the statement ‘x is a member of y if and only if, for all objects x, the general predication of x is identical to y, where x is a member of the domain of y.’ The expression x ∈ 𝔇y makes this definition well-defined by ensuring that x is a bound variable in its predication of membership in y.

In this case, the domain of Px, which is the set containing all dependent logical values x that satisfy the stated conditions for membership in y, is called the Universe (U) of y. The range of Px, which is the set of all possible dependent set variables y resulting from satisfaction of the conditions of membership for x, is the power set of U such that the binary relation of the membership of x in y is any subset of the cartesian product U × 𝒫(U) (the Cartesian Product of set U with the Power Set of U).

---

Notation and terminology The binary relation "is an element of", also called set membership, is denoted by the symbol "∈". Writing

$x\; \backslash in\; A$

means that " x is an element of  A". Equivalent expressions are " x is a member of  A", " x belongs to  A", " x is in  A" and " x lies in  A". The expressions " A includes x" and " A contains x" are also used to mean set membership, although some authors use them to mean instead " x is a subset of  A".

p. 12 Logician George Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.

For the relation ∈ , the converse relation ∈^T may be written

$A\; \backslash ni\; x$

meaning " A contains or includes x".

The negation of set membership is denoted by the symbol "∉". Writing

$x\; \backslash notin\; A$

means that " x is not an element of  A".

The symbol ∈ was first used by Giuseppe Peano, in his 1889 work italic=yes. Here he wrote on page X:

Signum significat est. Ita legitur a est quoddam b; …

which means

The symbol ∈ means is. So is read as a is a certain b; …

The symbol itself is a stylized lowercase Greek letter epsilon ("ϵ"), the first letter of the word , which means "is".

---

Examples Using the sets defined above, namely A = {1, 2, 3, 4}, B = {1, 2, {3, 4}} and C = {red, 12, B}, the following statements are true:

---

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 set B and set C are both 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 positive integers .

---

Formal relation As a relation, set membership must have a domain and a range. Conventionally the domain is called the universe denoted U. The range is the set of of U called the power set of U and denoted P( U). Thus the relation $\backslash in$ is a subset of . The converse relation $\backslash ni$ is a subset of .

---

See also

• Identity element
• Singleton (mathematics)

---

Further reading

• - "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).
• - 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".

Post Comment