Symbols:Set Theory

Set Delimiters
$$\left\{{x, y, z}\right\}$$

Denotes that the objects $$x, y, z$$ are the elements of a set.

The LaTeX code for this is \left\{{x, y, z}\right\}.

Empty Set
$$\varnothing$$

The empty set: $$\varnothing = \{\}$$.

An alternative but less attractive symbol for the same thing is $$\O$$.

The LaTeX code for $$\varnothing$$ is \varnothing, and for $$\O$$ it is \emptyset or \O.

Set Intersection
$$\cap$$

"Set Intersection". $$S \cap T$$ is the set containing all the elements that are in both the sets $$S$$ and $$T$$. $$S \cap T = \left\{ {x: x \in S \wedge x \in T}\right\}$$.

Its LaTeX code is \cap.

Set Union
$$\cup$$

"Set Union". $$S \cup T$$ is the set containing all the elements that are in either or both of the sets $$S$$ and $$T$$. $$S \cup T = \left\{ {x: x \in S \vee x \in T}\right\}$$.

Its LaTeX code is \cup.

Ordered Sum
$$+$$

$$S_1 + S_2$$ denotes the ordered sum of two sets $$S_1$$ and $$S_2$$.

See Arithmetic and Algebra and Abstract Algebra for alternative definitions of this symbol.

Its LaTeX code is +.

Set Difference
$$-$$

The difference between two sets $$S$$ and $$T$$ is denoted $$S - T$$ and consists of all the elements of $$S$$ which are not elements of $$T$$, that is, $$S - T = \left\{{x \in S: x \notin T}\right\}$$.

Its LaTeX code is -.

See Arithmetic and Algebra and Logical Operators for alternative definitions of this symbol.

Cartesian Product
$$\times$$

The Cartesian product.

Its LaTeX code is \times</tt>.

See Arithmetic and Algebra and Vector Algebra for alternative definitions of this symbol.

Is an Element Of
$$\in$$

"Element of". $$x \in S$$ means that $$x$$ is an element (or part) of the set $$S$$.

Its LaTeX code is \in</tt>.

Universal Quantifier
$$\forall$$

"For all".


 * $$\forall x \in S: P(x)$$ means that the propositional function $$P(x)$$ is true for every $$x$$ in the set $$S$$.


 * $$\forall x: P(x)$$ means that the propositional function $$P(x)$$ is true for every $$x$$ in the universal set.

Its LaTeX code is \forall</tt>.

Existential Quantifier
$$\exists$$

"There exists".


 * $$\exists x \in S: P(x)$$ means that there exists at least one $$x$$ in the set $$S$$ for which the propositional function $$P(x)$$ is true.


 * $$\exists x: P(x)$$ means that there exists at least one $$x$$ in the universal set for which the propositional function $$P(x)$$ is true.

Its LaTeX code is \exists</tt>.

Cardinality
$$\left|{S}\right|$$

The cardinality of the set $$S$$.

For finite sets, this means the number of elements in $$S$$.

The LaTeX code for this is \left|{S}\right|</tt>.

See Arithmetic and Algebra, Complex Analysis and Abstract Algebra for alternative definitions of this symbol.

Subset
$$\subseteq$$

"Subset". $$S \subseteq T$$ means "$$S$$ is a subset of $$T$$". In other words, every element of $$S$$ is also an element of $$T$$. Note that this symbol allows the possibility that $$S = T$$.

Its LaTeX code is \subseteq</tt>.

Proper Subset
$$\subset$$ or $$\subsetneq$$

"Proper subset". $$S \subset T$$ means "$$S$$ is a proper subset of $$T$$", in other words, $$S \subseteq T$$ and $$S \neq T$$. The symbols $$\subset$$ and $$\subsetneq$$ are equivalent.

The LaTeX code for $$\subset$$ is \subset</tt> and the LaTeX code for $$\subsetneq$$ is \subsetneq</tt>.

Superset
$$\supseteq$$

"Superset". $$S \supseteq T$$ means "$$S$$ is a superset of $$T$$", or equivalently, "$$T$$ is a subset of $$S$$". Thus every element of $$T$$ is also an element of $$S$$. Note that this symbol allows the possibility that $$S = T$$.

Its LaTeX code is \supseteq</tt>.

Proper Superset
$$\supset$$ or $$\supsetneq$$

"Proper superset". $$S \supset T$$ means $$S$$ is a proper superset of $$T$$, in other words, $$S \supseteq T$$ and $$S \neq T$$. The symbols $$\supset$$ and $$\supsetneq$$ are equivalent.

The LaTeX code for $$\supset$$ is \supset</tt> and the LaTeX code for $$\supsetneq$$ is \supsetneq</tt>.

Negation
$$\not\in, \not\exists, \not\subseteq, \not\subset, \not\supseteq, \not\supset$$

"Negation". The above symbols all mean the opposite of the non struck through version of the symbol. For example, $$x \not\in S$$ means that $$x$$ is not an element of $$S$$. The slash through a symbol ($$/$$ ) can be used to reverse the meaning of essentially any mathematical symbol (especially relations), although it is used most frequently with those listed above. Note that $$\not\subsetneq$$ and $$\not\supsetneq$$ can be confusing due to the strike through of the symbol as a whole and the strike through of the equivalence bar on the bottom, and hence should likely be avoided.

The LaTeX code for negation is \not</tt> followed by the code for whatever symbol you want to negate. For example, \not \in</tt> will render $$\not\in$$.

= Alternative Symbols =

Set Difference
$$\setminus$$

An alternative notation for the difference between two sets $$S$$ and $$T$$ is $$S \setminus T$$.

Its LaTeX code is \setminus</tt>.

See Number Theory: Divisor for an alternative use of this symbol.

= Deprecated Symbols =

Subset, Superset
$$\subset$$ is sometimes used to mean "$$S$$ is a subset of $$T$$" in the sense that $$S$$ is permitted to equal $$T$$, that is, for which we have specified as $$S \subseteq T$$.

Similarly, $$\supset$$ is sometimes used to mean $$S \supseteq T$$.

Although many sources use these interpretations, they are emphatically not recommended, as they can be the cause of considerable confusion.