$$x \in X$$ = x is in X

$$X \ni x$$ = X contains x (subtly different from the above)

$$x \notin X$$ = x is not in X

$$Y \subseteq X$$ = Y is a subset of or is equal to X

$$ Y \subset X $$ = Y is a proper subset of X i.e. not equal to X

$$ f: X \to Y$$ = the function f maps elements in X to elements in Y

$$ f\;\circ \; g$$ = function composition, do g

*first*, and

*then*do f

$$\exist x $$ = there is some x

$$\forall x$$ = for all x

$$\emptyset$$ = the empty set

$$X \cap Y$$ = X intersect Y i.e. the elements that X and Y share

$$X \cup Y$$ = the union of X and Y i.e. the set that is all elements of X and all elements of Y in no particular order

$$X \times Y$$ = Cartesian product i.e. the set whose elements are the ordered pair (x, y), x in X, y in Y

The following equalities may also be of some use;

$$X \cap \emptyset = \emptyset$$ always

$$ X \cup \emptyset = X$$ always

$$ X \cap X = X$$ always

$$ X \cup X = X$$ always

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