Given two nonempty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e.P×Q={{((p,q)):p∈P, q∈Q }}$P\times Q=\{(p,q):p\in P,\phantom{\rule{0.22em}{0ex}}q\in Q\phantom{\rule{0.22em}{0ex}}\}$ P×Q ≠Q×P $P\times Q\phantom{\rule{0.22em}{0ex}}\ne Q\times P\phantom{\rule{0.22em}{0ex}}$ 
A relation R from a nonempty set A to a nonempty set B is a subset ofthe cartesian product A×B $A\times B\phantom{\rule{0.22em}{0ex}}$. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A×B$A\times B$. The second element is called the image of the first element. 
The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R. 
The set of all second elements in a relation R from a set A to a set B iscalled the range of the relation R. The whole set B is called the codomain of the relation R. range ⊂ codomain. 
The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A×B$A\times B$. If n((A )) = p and n((B)) = q, thenn ((A×B))=pq$(A\times B)=pq$ and the total number of relations is 2pq $2}^{pq}\phantom{\rule{0.22em}{0ex}$ 
Relation=If ((a,b))∈R $Relation=If\phantom{\rule{0.22em}{0ex}}(a,b)\in R\phantom{\rule{0.22em}{0ex}}$ we say that a is related to b under relation R. It is written as a R b 
A relation R in a set A is called empty relation, if no element of A is related to any element of AR=𝜙⊂A⨯A $R=\mathit{\phi}\subset A\u2a2fA\phantom{\rule{0.22em}{0ex}}$ 
A relation R in a set A is called universal relation, if each element of A is related to every element of AR=A⨯A $R=A\u2a2fA\phantom{\rule{0.22em}{0ex}}$ 
Reflexive: A relation R in a set A is called reflexive if ((a,a))∈R$(a,a)\in R$, for every a∈R$a\in R$ 
Symmetric: A relation R in a set A is called symmetric if ((a1,a2))∈R$({a}_{1},{a}_{2})\in R$ implies that ((a2,a1))∈R$({a}_{2},{a}_{1})\in R$ for all a1,a2∈R${a}_{1},{a}_{2}\in R$ 
Transitive: A relation R in a set A is called symmetric if ((a1,a2))∈R$({a}_{1},{a}_{2})\in R$ and ((a2,a3))∈R$({a}_{2},{a}_{3})\in R$ implies that ((a1,a3))∈R$({a}_{1},{a}_{3})\in R$ for all a1,a2, a3∈R${a}_{1},{a}_{2},\phantom{\rule{0.22em}{0ex}}{a}_{3}\in R$ 
Equivalence:A relation R in a set A is called equivalence relation, if R is reflexive, symmetric and transitive. 
FunctionA relation "f" from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. 
Identity Function: Let R be the set of real numbers.If a real valued function is defined as f:R→R $f:R\to R\phantom{\rule{0.22em}{0ex}}$ $}$by y=f((x))=x $y=f(x)=x\phantom{\rule{0.22em}{0ex}}$ for each x∈R$x\in R$ then it is called identity function. 
Constant Function: Let R be the set of real numbers.If a real valued function is defined as f:R→R $f:R\to R\phantom{\rule{0.22em}{0ex}}$ $}$by y=f((x))=c $y=f(x)=c\phantom{\rule{0.22em}{0ex}}$ for each x∈R$x\in R$ where c is contant, then it is called identity function. 
Addition of two functionsLet f : X →R$f\phantom{\rule{0.22em}{0ex}}:\phantom{\rule{0.22em}{0ex}}X\phantom{\rule{0.22em}{0ex}}\to R$ and g: X →R$\phantom{\rule{0.22em}{0ex}}g:\phantom{\rule{0.22em}{0ex}}X\phantom{\rule{0.22em}{0ex}}\to R$ be any two real functions, where X⊂R $X\subset R\phantom{\rule{0.22em}{0ex}}$. Then, we define ((f+g)) : X →R$(f+g)\phantom{\rule{0.22em}{0ex}}:\phantom{\rule{0.22em}{0ex}}X\phantom{\rule{0.22em}{0ex}}\to R$ by((f+g))((x))=f((x))+g((x)) $(f+g)(x)=f(x)+g(x)\phantom{\rule{0.22em}{0ex}}$, for all x∈X$x\in X$ 
Subtraction of two functionsLet f : X →R$f\phantom{\rule{0.22em}{0ex}}:\phantom{\rule{0.22em}{0ex}}X\phantom{\rule{0.22em}{0ex}}\to R$ and g: X →R$\phantom{\rule{0.22em}{0ex}}g:\phantom{\rule{0.22em}{0ex}}X\phantom{\rule{0.22em}{0ex}}\to R$ be any two real functions, where X⊂R $X\subset R\phantom{\rule{0.22em}{0ex}}$. Then, we define ((fg)) : X →R$(fg)\phantom{\rule{0.22em}{0ex}}:\phantom{\rule{0.22em}{0ex}}X\phantom{\rule{0.22em}{0ex}}\to R$ by((fg))((x))=f((x))g((x)) $(fg)(x)=f(x)g(x)\phantom{\rule{0.22em}{0ex}}$, for all x∈X$x\in X$ 

OneOne (or injective)A function f:X→Y $f:X\to Y\phantom{\rule{0.22em}{0ex}}$ is defined to be oneone (or injective) if the images of the distinct elements of X dunder "f" are distinct. It means that afor every x1,x2∈X , if f((x1))=f((x2)) implies that, x1=x2$\begin{array}{l}for\phantom{\rule{0.22em}{0ex}}every\phantom{\rule{0.44em}{0ex}}{x}_{1},{x}_{2}\in X\phantom{\rule{0.44em}{0ex}},\phantom{\rule{0.44em}{0ex}}\\ if\phantom{\rule{0.22em}{0ex}}f({x}_{1})=f({x}_{2})\phantom{\rule{0.22em}{0ex}}\\ implies\phantom{\rule{0.22em}{0ex}}that,\phantom{\rule{0.22em}{0ex}}{x}_{1}={x}_{2}\end{array}$ 
Onto (or surjective)A function f:X→Y is defined to be onto ((or surjective))$f:X\to Y\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}defined\phantom{\rule{0.22em}{0ex}}to\phantom{\rule{0.22em}{0ex}}be\phantom{\rule{0.22em}{0ex}}onto\phantom{\rule{0.22em}{0ex}}(or\phantom{\rule{0.22em}{0ex}}surjective)$ if every element of Y is the image of some element of X under "f".It means for every y∈Y$y\in Y$ there exist an element x in X such that f(x)=y 
oneone and onto (bijective)function f:X→Y $f:X\to Y\phantom{\rule{0.22em}{0ex}}$ is said to be oneone and onto (bijective), if "f" is both oneone and onto. 
Composition of function"Composition of Function" is applying one function to the results of another. 
Identity function, is a function that always returns the same value that was used as its argument.Example: f(x)=x 
A function f:X→Y $f:X\to Y\phantom{\rule{0.22em}{0ex}}$ is defined to be invertible, if there exists a function g:Y→X$g:Y\to X$ such that gof(x) = x and fog(x) = x. The function g is called the inverse of f and is denoted by f1$f}^{1$ 
If f:X→Y , g:Y→Z and h:Z→S$f:X\to Y\phantom{\rule{0.44em}{0ex}},\phantom{\rule{0.44em}{0ex}}g:Y\to Z\phantom{\rule{0.44em}{0ex}}and\phantom{\rule{0.44em}{0ex}}h:Z\to S$ , are functions, then ho((gof)) = ((hog))of $ho(gof)\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}(hog)of\phantom{\rule{0.22em}{0ex}}$ 