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A set where all its elements are contained within another set. Notation: $A \subseteq B$.

The set of all distinct elements that are in either set. Notation: $A \cup B$.

The set of elements that are in one set but not in the other. Notation: $A - B$ or $A \setminus B$.

The set of elements which are in either of the sets and not in their intersection. Notation: $A \triangle B$.

The set that contains all possible outputs of a function. Notation: If $f: A \to B$, then $B$ is the codomain.

A function where each element of the range is mapped to by exactly one element of the domain. Notation: $f$ is injective if $f(a) = f(b)$ implies $a = b$.

The number of elements in a set. Notation: $|A|$ or $\#A$.

A set with no elements. Notation: $\emptyset$ or $\{\}$.

The set of all elements in the universal set that are not in the given set. Notation: $A^c$ or $\bar{A}$.

A division of a set into non-overlapping subsets that cover all elements of the original set.

The set that contains all objects or elements under consideration. Notation: Usually denoted by $U$ or $\xi$.

Two sets with no common elements. Notation: $A \cap B = \emptyset$.

A relation that is reflexive, symmetric, and transitive. Notation: $a \sim b$ indicates $a$ is equivalent to $b$.

A function where the output value is uniquely determined by the input, adhering to the definition of a function.

A pair of elements with an order, where the first element is considered the first component of the pair. Notation: $(a, b)$.

The set of all possible inputs for which the function is defined. Notation: If $f: A \to B$, then $A$ is the domain.

The set of all subsets of a set, including the empty set and the set itself. Notation: $\mathcal{P}(A)$ or $2^A$.

The set of all ordered pairs obtained by the product of two sets. Notation: $A \times B$.

A relation between sets that associates each element of a set with exactly one element of another set. Notation: $f: A \to B$.

A function that is both injective and surjective, meaning it is a one-to-one correspondence between the domain and codomain. Notation: $f$ is bijective if it is both one-to-one and onto.

A function that reverses the direction of a given function, if such a function exists. Notation: $f^{-1}$ for the inverse of $f$.

A function where each element of the codomain is the image of at least one element of the domain. Notation: $f$ is surjective if for every $y$ in the codomain, there is an $x$ in the domain such that $f(x) = y$.

The set of elements that are common to both sets. Notation: $A \cap B$.

A subset that is not equal to the parent set. Notation: $A \subsetneq B$.