Symbols:Set Theory

$$\cap$$

"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".

$$\cup$$

"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".

$$+$$

$$S_1 + S_2$$ 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 "+".

$$-$$

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, $$S - T = \left\{{x \in S: x \notin T}\right\}$$.

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

Its LaTeX code is "-".

$$\times$$

The Cartesian product.

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

Its LaTeX code is "\times".

$$\in$$

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

Its LaTeX code is "\in".

$$\forall$$

"For all". $$\forall x:$$ means that the following statement is true for every $$x$$ in the universal set, or for every $$x$$ in the set $$A$$ in the case of $$\forall x \in A$$.

Its LaTeX code is "\forall".

$$\exists$$

"There exists". $$\exists x:$$ means that there is at least one $$x$$ in the universal set for which the following statement holds, or there exists at least one $$x$$ in $$A$$ in the case of $$\exists x \in A$$.

Its LaTeX code is "\exists".

$$\subseteq$$

"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".

$$\subset$$ 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" and the LaTeX code for $$\subsetneq$$ is "\subsetneq".

$$\supseteq$$

"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".

$$\supset$$ 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" and the LaTeX code for $$\supsetneq$$ is "\supsetneq".

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

"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 followed by the code for whatever symbol you want to negate. For example, "\not\in" will render $$\not\in$$.