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):pP,qQ} P×QQ×P
A relation R from a non-empty set A to a non-empty set B is a subset ofthe cartesian product A×B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in 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. If n((A )) = p and n((B)) = q, thenn (A×B)=pq and the total number of relations is 2pq
Relation=If(a,b)R 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=𝜙AA
A relation R in a set A is called universal relation, if each element of A is related to every element of AR=AA
Reflexive: A relation R in a set A is called reflexive if (a,a)R, for every aR
Symmetric: A relation R in a set A is called symmetric if (a1,a2)R implies that (a2,a1)R for all a1,a2R
Transitive: A relation R in a set A is called symmetric if (a1,a2)R and (a2,a3)R implies that (a1,a3)R for all a1,a2,a3R
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:RR by y=f(x)=x for each xR 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:RR by y=f(x)=c for each xR where c is contant, then it is called identity function.
Addition of two functionsLet f:XR and g:XR be any two real functions, where XR. Then, we define (f+g):XR by(f+g)(x)=f(x)+g(x), for all xX
Subtraction of two functionsLet f:XR and g:XR be any two real functions, where XR. Then, we define (f-g):XR by(f-g)(x)=f(x)-g(x), for all xX
One-One (or injective)A function f:XY is defined to be one-one (or injective) if the images of the distinct elements of X dunder "f" are distinct. It means that foreveryx1,x2X,iff(x1)=f(x2)impliesthat,x1=x2
Onto (or surjective)A function f:XYisdefinedtobeonto(orsurjective) if every element of Y is the image of some element of X under "f".It means for every yY there exist an element x in X such that f(x)=y
one-one and onto (bijective)function f:XY 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:XY is defined to be invertible, if there exists a function g:YX such that gof(x) = x and fog(x) = x. The function g is called the inverse of f and is denoted by f-1
If f:XY,g:YZandh:ZS , are functions, then ho(gof)=(hog)of
Q.2.5_5IfA={-1,1},findA×A×A A.2.5_5A×A×A={-1,1}×{-1,1}×{-1,1}A×A×A={(-1,-1,-1),(-1,-1,1),(-1,1,-1),(-1,1,1),(1,-1,-1),(1,-1,1),(1,1,-1),(1,1,1)} Q.2.3_2 Find the domain and range of the following real functions:i)f(x)=-|x|ii)f(x)=9-x2 A.2.3_2 i)f(x)=-|x|, 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)f(x)=9-x2, This is the real value function.x can be any real number.9-x2 cannot be negative, so9-x20(3-x)(3+x)0Soxmustbebetween-3and+3,i.e[-3,+3]Atx=0,f(0)=3Atx=3,f(3)=0So domain: [-3,+3].Range: [0,3]