Set Theory

2024-06-1400:26 Status:Incomplete Tags:

∅

Denotes the empty set, and is more often written ∅. Using set-builder notation, it may also be denoted {}.

Number of elements: $# S$ may denote the cardinality of the set S. An alternative notation is $∣ S ∣$ .
Primorial: $# n$ denotes the product of the prime numbers that are not greater than $n$ .
In topology, $M # N$ denotes the connected sum of two manifolds or two knots.

∈

Denotes set membership, and is read “is in” or “belongs to”. That is, $x \in S$ means that x is an element of the set S.

∉

⊂

Denotes set inclusion. However two slightly different definitions are common.
$A \subset B$ may mean that A is a subset of B, and is possibly equal to B; that is, every element of A belongs to B; in formula, $\forall x, x \in A \Rightarrow x \in B$ .
$A \subset B$ may mean that A is a proper subset of B, that is the two sets are different, and every element of A belongs to B; in formula, $A \neq = B \land \forall x, x \in A \Rightarrow x \in B$ .

⊆

$A \subseteq B$ means that A is a subset of B. Used for emphasizing that equality is possible, or when means that when $A \subset B$ , $A$ is a proper subset of $B$ .

⊊

$A ⊊ B$ means that A is a proper subset of B. Used for emphasizing that $A \neq = B$ , or when $A \subset B$ does not imply that $A$ is a proper subset of $B$ .

⊃, ⊇, ⊋

Denote the converse relation of ⊂, ⊆, ⊊ respectively. For example, $A \subset B$ is equivalent to $B \supset A$ .

∪

Denotes set-theoretic union, that is, $A \cup B$ is the set formed by the elements of $A$ and $B$ together. That is, $A \cup B = x ∣ (x \in A) \lor (x \in B)$

∩

Denotes set-theoretic intersection, that is, $A \cap B$ is the set formed by the elements of both $A$ and $B$ . That is, $A \cup B = x ∣ (x \in A) \land (x \in B)$

Set difference; that is, $A ∖ B$ is the set formed by the elements of $A$ that are not in $B$ . Sometimes, $A - B$ ! is used instead; see – in § Arithmetic operators.

⊖ or △

Symmetric difference: that is, $A ⊖ B$ or $A Δ B$ is the set formed by the elements that belong to exactly one of the two sets $A$ and $B$ .

∁

With a subscript, denotes a set complement: that is, if $B \subseteq A$ then $∁_{A} B = A ∖ B$ .
Without a subscript, denotes the absolute complement; that is, $∁ A = ∁_{U} A$ , where U is a set implicitly defined by the context, which contains all sets under consideration. This set U is sometimes called the universe of discourse.

See also × in § Arithmetic operators.
Denotes the Cartesian product of two sets. That is, $A \times B$ is the set formed by all pairs of an element of $A$ and an element of $B$ .
Denotes the direct product of two mathematical structures of the same type, which is the Cartesian product of the underlying sets, equipped with a structure of the same type. For example, direct product of rings, direct product of topological spaces
In category theory, denotes the direct product (often called simply product) of two objects, which is a generalization of the preceding concepts of product.

⊔

Denotes the disjoint union. That is, if $A$ and $B$ are sets then $A ⊔ B = (A \times i_{A}) \cup (B \times i_{B})$ is a set of pairs where $i_{A}$ and $i_{B}$ are distinct indices discriminating the members of $A$ and $B$ in $A⊔B.

⨆ or ∐

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