Relations and Functions

Vocabulary practice flashcards covering fundamental definitions, classifications, and concepts of relations, functions, equivalence relations, composition, and invertibility.

Relation

A mathematical link from set $A$ to set $B$, defined formally as an arbitrary subset of the Cartesian product $A \times B$.

Empty Relation

A relation $R$ in a set $A$ in which no element of $A$ is related to any element of $A$, denoted by $R = \phi \subset A \times A$.

Universal Relation

A relation $R$ in a set $A$ in which each element of $A$ is related to every element of $A$, denoted by $R = A \times A$.

Trivial Relations

The term used to refer collectively to both the empty relation and the universal relation in a set.

Reflexive Relation

A relation $R$ in a set $A$ where $(a, a) \in R$ for every $a \in A$.

Symmetric Relation

A relation $R$ in a set $A$ where $(a_1, a_2) \in R$ implies that $(a_2, a_1) \in R$ for all $a_1, a_2 \in A$.

Transitive Relation

A relation $R$ in a set $A$ where $(a_1, a_2) \in R$ and $(a_2, a_3) \in R$ implies that $(a_1, a_3) \in R$ for all $a_1, a_2, a_3 \in A$.

Equivalence Relation

A relation $R$ in a set $A$ that is reflexive, symmetric, and transitive.

Equivalence Class

For an equivalence relation $R$ in a set $X$, the subset denoted by $[a]$ containing all elements of $X$ that are related to the element $a$.

One-One Function (Injective)

A function $f : X \rightarrow Y$ where the images of distinct elements of $X$ under $f$ are distinct, such that $f(x_1) = f(x_2)$ implies $x_1 = x_2$ for every $x_1, x_2 \in X$.

Many-One Function

A function $f : X \rightarrow Y$ that is not one-one, where two or more distinct elements in the domain have the same image in the co-domain.

Onto Function (Surjective)

A function $f : X \rightarrow Y$ where every element of $Y$ is the image of some element of $X$ under $f$, meaning that for every $y \in Y$, there exists an element $x \in X$ such that $f(x) = y$, or equivalently, the range of $f$ equals $Y$.

Bijective Function

A function $f : X \rightarrow Y$ that is both one-one (injective) and onto (surjective).

Composition of Functions

For two functions $f : A \rightarrow B$ and $g : B \rightarrow C$, the combined function denoted by $g \circ f : A \rightarrow C$ defined by $(g \circ f)(x) = g(f(x))$ for all $x \in A$.

Invertible Function

A function $f : X \rightarrow Y$ for which there exists a function $g : Y \rightarrow X$ such that $g \circ f = I_X$ and $f \circ g = I_Y$.

Inverse Function

The unique function denoted by $f^{-1} : Y \rightarrow X$ associated with an invertible function $f : X \rightarrow Y$ that satisfies $f^{-1} \circ f = I_X$ and $f \circ f^{-1} = I_Y$.

Equal Functions

Two functions $f : A \rightarrow B$ and $g : A \rightarrow B$ that satisfy $f(a) = g(a)$ for every element $a \in A$.

Lejeune Dirichlet

The mathematician ($1805$-$1859$) who formulated the definition of function used prior to the introduction of the modern set-theoretic definition.