Relations and Functions

Vocabulary terms and definitions covering the fundamental concepts of relations, types of relations, types of functions, and function compositions as presented in the lecture notes.

Relation

Mathematically defined as any arbitrary subset of the Cartesian product $A \times B$. If $(a, b) \times R$, it is said that $a$ is related to $b$ under the relation $R$, written as $a R b$.

Empty relation

A relation $R$ in a set $A$ where no element of $A$ is related to any element of $A$, such that $$R =                                     

                 

                      

  

     

               

            

    

     

        

  

     

               

                 



  

    

     

     

  

      



  

         

      

    

 

     

          

 

     

           

  

         

         

        

  

     

      

      

 

 

    

       



 

Universal relation

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

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 $A$ satisfying the condition that if $(a_1, a_2) \in R$, then $(a_2, a_1) \in R$, for all $a_1, a_2 \in A$.

Transitive relation

A relation $R$ in a set $A$ satisfying the condition that if $(a_1, a_2) \in R$ and $(a_2, a_3) \in R$, then $(a_1, a_3) \in R$, for all $a_1, a_2, a_3 \in A$.

Equivalence relation

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

Equivalence class

Denoted by $[a]$, it is the subset of a set $X$ containing all elements $b$ that are related to an element $a \in X$ under an equivalence relation $R$. Every equivalence relation partitions the set into mutually disjoint equivalence classes.

One-one (Injective) function

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

Many-one

A function that is not one-one, meaning at least two distinct elements in the domain have the same image in the co-domain.

Onto (Surjective) function

A function $f : X \rightarrow Y$ where every element of $Y$ is the image of at least one element of $X$ under $f$. This occurs if and only if the Range of $f = Y$.

One-one and onto (Bijective)

A function that is both injective (one-one) and surjective (onto).

Composition of functions

Given functions $f : A \rightarrow B$ and $g : B \rightarrow C$, the composition $gof : A \rightarrow C$ is defined as $gof(x) = g(f(x))$ 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 $gof = I_X$ and $fog = I_Y$. A function is invertible if and only if it is bijective.

Greatest Integer Function

A function $f : R \rightarrow R$ defined by $f(x) = [x]$, which denotes the greatest integer less than or equal to $x$. It is proved to be neither one-one nor onto.

Modulus Function

A function $f : R \rightarrow R$ defined by $f(x) = |x|$, where $|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$. He is neither one-one nor onto.

Signum Function

A function $f : R \rightarrow R$ defined as $f(x) = 1$ if $x > 0$, $f(x) = 0$ if $x = 0$, and $f(x) = -1$ if $x < 0$. It is neither one-one nor onto.

Lejeune Dirichlet

A mathematician (1805-1859) whose definition of a function was used until the development of the set theoretic definition.