Relations and Functions Vocabulary

Flashcards on core mathematical concepts, relation types, and function definitions from Class 12 Relations and Functions.

Relation

A subset of $A \times B$ (denoted $R \subseteq A \times B$) from a set $A$ to a set $B$. A relation from a set $A$ to itself is called a relation in $A$.

Empty Relation

A relation $R$ in a set $A$ where no element of $A$ is related to any element of $A$, denoted as $R = \phi \subset A \times A$. Also called a void relation.

Universal Relation

A relation $R$ in a set $A$ where each element of $A$ is related to every element of $A$, expressed as $R = A \times A$.

Reflexive Relation

A relation $R$ in a set $A$ where every element is related to itself, i.e., $(a, a) \in R$ for all $a \in A$.

Symmetric Relation

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

Transitive Relation

A relation $R$ in a set $A$ where $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$ for all $a, b, c \in A$.

Equivalence Relation

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

Equivalence Class

In a relation $R$ on a set $A$, the set of all elements related to a given element $a \in A$, denoted by $[a] = \{x \in A : (x, a) \in R\}$.

Inverse Relation

The relation $R^{-1}$ defined as the inverse relation of $R$ if $(a, b) \in R \iff (b, a) \in R^{-1}$, where $\text{Domain}(R) = \text{Range}(R^{-1})$ and $\text{Range}(R) = \text{Domain}(R^{-1})$.

Identity Relation

A relation $R$ in a set $A$ defined as $R = \{(a, a) \mid \forall a \in A\}$.

Function

A relation $f$ from a set $A$ to a set $B$ where every element of set $A$ has only one image in set $B$.

One-one (Injective) Function

A function $f: A \rightarrow B$ where distinct elements of $A$ have distinct images in $B$ (i.e., $f(a) = f(b) \implies a = b$), satisfying $n(A) \leq n(B)$.

Many-one Function

A function $f: A \rightarrow B$ where two or more elements of $A$ have the same image in $B$ (i.e., for $a, b \in A$ such that $a \neq b$, $f(a) = f(b)$).

Onto (Surjective) Function

A function $f: A \rightarrow B$ where every element of $B$ has at least one pre-image in $A$, satisfying $n(A) \geq n(B)$ and $\text{Range} = \text{Codomain}$.

Into Function

A function $f: A \rightarrow B$ where there exists at least one single element in $B$ having no pre-image in $A$.

Bijective Function

A function $f: A \rightarrow B$ that is both one-one and onto, satisfying $n(A) = n(B)$ and $\text{Range} = \text{Codomain}$.

Horizontal Line Test for One-One Function

A graphical method stating that if no horizontal line intersects the graph of a function in more than one point, then the function is a one-one function.

Horizontal Line Test for Many-One Function

A graphical method stating that if a line parallel to the X-axis (horizontal line) intersects the graph of a function in more than one point, then the function is a many-one function.