Symbols:Set Theory

$$\cap$$

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

$$\cup$$

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

$$+$$

$$S_1 + S_2$$ 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.

$$-$$

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.

$$\times$$

The Cartesian product.

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

$$\in$$

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

$$\forall$$

"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 $$\forall x \in A$$.

$$\exists$$

"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 $$\exists x \in A$$.

$$\subseteq$$

"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 $$S = T$$.

$$\subset$$ 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.

$$\supseteq$$

"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 $$S = T$$.

$$\supset$$ 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.

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

"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.