math_relation_functionRelation and Function
 Given two non-empty 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 }}$P×Q=\left\{\left(p,q\right):p\in P,\phantom{\rule{0.22em}{0ex}}q\in Q\phantom{\rule{0.22em}{0ex}}\right\}$ P×Q ≠Q×P $P×Q\phantom{\rule{0.22em}{0ex}}\ne Q×P\phantom{\rule{0.22em}{0ex}}$ A relation R from a non-empty set A to a non-empty set B is a subset ofthe cartesian product A×B $A×B\phantom{\rule{0.22em}{0ex}}$. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A×B$A×B$. 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$A×B$. If n((A )) = p and n((B)) = q, thenn ((A×B))=pq$\left(A×B\right)=pq$ and the total number of relations is 2pq ${2}^{pq}\phantom{\rule{0.22em}{0ex}}$ Relation=If ((a,b))∈R $Relation=If\phantom{\rule{0.22em}{0ex}}\left(a,b\right)\in R\phantom{\rule{0.22em}{0ex}}$ 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 $R=𝜙\subset A⨯A\phantom{\rule{0.22em}{0ex}}$ 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 $R=A⨯A\phantom{\rule{0.22em}{0ex}}$ Reflexive: A relation R in a set A is called reflexive if ((a,a))∈R$\left(a,a\right)\in R$, for every a∈R$a\in R$ Symmetric: A relation R in a set A is called symmetric if ((a1,a2))∈R$\left({a}_{1},{a}_{2}\right)\in R$ implies that ((a2,a1))∈R$\left({a}_{2},{a}_{1}\right)\in R$ 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$\left({a}_{1},{a}_{2}\right)\in R$ and ((a2,a3))∈R$\left({a}_{2},{a}_{3}\right)\in R$ implies that ((a1,a3))∈R$\left({a}_{1},{a}_{3}\right)\in R$ 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 $y=f\left(x\right)=x\phantom{\rule{0.22em}{0ex}}$ 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 $y=f\left(x\right)=c\phantom{\rule{0.22em}{0ex}}$ 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$\left(f+g\right)\phantom{\rule{0.22em}{0ex}}:\phantom{\rule{0.22em}{0ex}}X\phantom{\rule{0.22em}{0ex}}\to R$ by((f+g))((x))=f((x))+g((x)) $\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\phantom{\rule{0.22em}{0ex}}$, 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 ((f-g)) : X →R$\left(f-g\right)\phantom{\rule{0.22em}{0ex}}:\phantom{\rule{0.22em}{0ex}}X\phantom{\rule{0.22em}{0ex}}\to R$ by((f-g))((x))=f((x))-g((x)) $\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\phantom{\rule{0.22em}{0ex}}$, for all x∈X$x\in X$ One-One (or injective)A function f:X→Y $f:X\to Y\phantom{\rule{0.22em}{0ex}}$ is defined to be one-one (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$\begin{array}{l}for\phantom{\rule{0.22em}{0ex}}every\phantom{\rule{0.44em}{0ex}}{x}_{1},{x}_{2}\in X\phantom{\rule{0.44em}{0ex}},\phantom{\rule{0.44em}{0ex}}\\ if\phantom{\rule{0.22em}{0ex}}f\left({x}_{1}\right)=f\left({x}_{2}\right)\phantom{\rule{0.22em}{0ex}}\\ implies\phantom{\rule{0.22em}{0ex}}that,\phantom{\rule{0.22em}{0ex}}{x}_{1}={x}_{2}\end{array}$ Onto (or surjective)A function f:X→Y is defined to be onto ((or surjective))$f:X\to Y\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}defined\phantom{\rule{0.22em}{0ex}}to\phantom{\rule{0.22em}{0ex}}be\phantom{\rule{0.22em}{0ex}}onto\phantom{\rule{0.22em}{0ex}}\left(or\phantom{\rule{0.22em}{0ex}}surjective\right)$ 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 one-one and onto (bijective)function f:X→Y $f:X\to Y\phantom{\rule{0.22em}{0ex}}$ is said to be one-one and onto (bijective), if "f" is both one-one 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 f-1${f}^{-1}$ 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 $ho\left(gof\right)\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}\left(hog\right)of\phantom{\rule{0.22em}{0ex}}$
Q.2.5_5$If\phantom{\rule{0.22em}{0ex}}A=\left\{-1,1\right\}\phantom{\rule{0.22em}{0ex}},\phantom{\rule{0.22em}{0ex}}find\phantom{\rule{0.22em}{0ex}}A×A×A\phantom{\rule{0.22em}{0ex}}$ A.2.5_5$A×A×A=\left\{-1,1\right\}×\left\{-1,1\right\}×\left\{-1,1\right\}\phantom{\rule{0.22em}{0ex}}$$\begin{array}{l}A×A×A=\left\{\left(-1,-1,-1\right),\left(-1,-1,1\right),\\ \phantom{\rule{5.72em}{0ex}}\left(-1,1,-1\right),\left(-1,1,1\right),\\ \phantom{\rule{5.72em}{0ex}}\left(1,-1,-1\right),\left(1,-1,1\right),\\ \phantom{\rule{5.72em}{0ex}}\left(1,1,-1\right),\left(1,1,1\right)\right\}\end{array}$ Q.2.3_2 Find the domain and range of the following real functions:$i\right)\phantom{\rule{0.22em}{0ex}}f\left(x\right)=-|x|\phantom{\rule{0.22em}{0ex}}$$ii\right)\phantom{\rule{0.22em}{0ex}}f\left(x\right)=\sqrt{9-{x}^{2}}\phantom{\rule{0.22em}{0ex}}$ A.2.3_2 $i\right)\phantom{\rule{0.22em}{0ex}}f\left(x\right)=-|x|\phantom{\rule{0.22em}{0ex}}$, This is the real value function.x can be any real number.-|x| can be any negative real number or 0 (zero).So domain: any real number.Range: any negative real number or 0 (zero). $ii\right)\phantom{\rule{0.22em}{0ex}}f\left(x\right)=\sqrt{9-{x}^{2}}\phantom{\rule{0.22em}{0ex}}$, This is the real value function.x can be any real number.$9-{x}^{2}$ cannot be negative, so$\begin{array}{l}9-{x}^{2}\phantom{\rule{0.22em}{0ex}}⩾0\\ ⟹\left(3-x\right)\left(3+x\right)⩾0\\ So\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}must\phantom{\rule{0.22em}{0ex}}be\phantom{\rule{0.22em}{0ex}}between\phantom{\rule{0.22em}{0ex}}-3\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}+3\phantom{\rule{0.44em}{0ex}},\phantom{\rule{0.22em}{0ex}}i.e\phantom{\rule{0.22em}{0ex}}\left[-3,+3\right]\end{array}$$\begin{array}{l}At\phantom{\rule{0.22em}{0ex}}x=0,\phantom{\rule{0.22em}{0ex}}f\left(0\right)=3\\ At\phantom{\rule{0.22em}{0ex}}x=3,\phantom{\rule{0.22em}{0ex}}f\left(3\right)=0\end{array}$So domain: $\left[-3,+3\right]$.Range: [0,3]