Ordered Pair: If a pair of elements is listed in a specific order, then such a pair is called ordered pair.
Cartesian Product of Two Sets: Let A and B be any two non-empty sets, then the set of all ordered pairs (a,b) for all a∈A and b∈B is called the Cartesian product of A and B. It is written as A⨯B.
A⨯B={(a,b): for all a∈A,b∈B}
Relation: Let A and B be two non empty sets. Then, a relation R from A to B is a subset of A⨯B i.e. R⊂A⨯B.
Empty Relation: If R=Ï•, then R is called the empty relation.
Universal Relation: If R=A⨯B, then R is called the universal relation.
Domain of the relation R, is the set of all first component of the ordered pairs which belong to R.
Range of the relation R is the set of all second components of the ordered pairs which belong to R.
Co-domain is the set of all elements of second set.
Number of Relations: Let A and B be any two non-empty finite sets containing p and q elements respectively i.e. n(A)=p, n(B)=q, then
Number of ordered pairs in A⨯B = pq
Total number of relations from A to B = $\displaystyle \small 2^{pq}$
Function: If X,Y are two non-empty sets then a subset of X⨯Y is called a function from X to Y iff for each x∈X, there exists a unique y∈Y such that (x,y)∈f.
Let f be a function from X to Y, then
Set X is called the Domain of the function f.
Set Y is called the Co-domain of the function f.
Set consisting of all the images of the elements of X is called the Range of function f.
Types of Functions:
Constant Function: f(x)=k
Identity Function: f(x)=x
Polynomial Function: $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{n}x^{n}$
Rational Function: $f(x)=\frac{p(x)}{q(x)}$
The Modulus Function: $f(x)=|x|=\left\{\begin{matrix} x & if & x\geq 0\\ -x & if & x< 0 \end{matrix}\right.$
Signum Function: $f(x)=\left\{\begin{matrix} 1 & when &x> 0 \\ 0 & when &x= 0 \\ -1 & when &x< 0 \end{matrix}\right.$
Greatest Integer Function: f(x)=[x]
Cartesian Product of Two Sets: Let A and B be any two non-empty sets, then the set of all ordered pairs (a,b) for all a∈A and b∈B is called the Cartesian product of A and B. It is written as A⨯B.
A⨯B={(a,b): for all a∈A,b∈B}
Relation: Let A and B be two non empty sets. Then, a relation R from A to B is a subset of A⨯B i.e. R⊂A⨯B.
Empty Relation: If R=Ï•, then R is called the empty relation.
Universal Relation: If R=A⨯B, then R is called the universal relation.
Domain of the relation R, is the set of all first component of the ordered pairs which belong to R.
Range of the relation R is the set of all second components of the ordered pairs which belong to R.
Co-domain is the set of all elements of second set.
Number of Relations: Let A and B be any two non-empty finite sets containing p and q elements respectively i.e. n(A)=p, n(B)=q, then
Number of ordered pairs in A⨯B = pq
Total number of relations from A to B = $\displaystyle \small 2^{pq}$
Function: If X,Y are two non-empty sets then a subset of X⨯Y is called a function from X to Y iff for each x∈X, there exists a unique y∈Y such that (x,y)∈f.
Let f be a function from X to Y, then
Set X is called the Domain of the function f.
Set Y is called the Co-domain of the function f.
Set consisting of all the images of the elements of X is called the Range of function f.
Types of Functions:
Constant Function: f(x)=k
Identity Function: f(x)=x
Polynomial Function: $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{n}x^{n}$
Rational Function: $f(x)=\frac{p(x)}{q(x)}$
The Modulus Function: $f(x)=|x|=\left\{\begin{matrix} x & if & x\geq 0\\ -x & if & x< 0 \end{matrix}\right.$
Signum Function: $f(x)=\left\{\begin{matrix} 1 & when &x> 0 \\ 0 & when &x= 0 \\ -1 & when &x< 0 \end{matrix}\right.$
Greatest Integer Function: f(x)=[x]
Properties of Cartesian Product of Sets
(i) A⨯B = B⨯A ⇔ A=B
(ii) A⨯B = A⨯C ⇔ B=C
(iii) A⨯(B∪C) = (A⨯B)∪(A⨯C)
(iv) A⨯(B∩C) = (A⨯B)∩(A⨯C)
(v) A⨯(B−C) = (A⨯B)−(A⨯C)
(vi) (A⨯B)∩(C⨯D) = (A∩C)⨯(B∩D)
(v) (A⨯B)∩(B⨯A) = (A∩B)⨯(B∩A)
(vi) If A⊂B and C⊂D, then A⨯C⊂B⨯D
Algebra of Real Functions
Addition
(f+g)(x)= f(x)+g(x)
Subtraction
(f-g)(x)= f(x)-g(x)
Multiplication by a scalar
(cf)(x) = c f(x)
Multiplication
(fg)(x)= f(x)g(x)
Quotient
$\left ( \frac{f}{g} \right )\left ( x \right )=\frac{f(x)}{g(x)}$
(i) A⨯B = B⨯A ⇔ A=B
(ii) A⨯B = A⨯C ⇔ B=C
(iii) A⨯(B∪C) = (A⨯B)∪(A⨯C)
(iv) A⨯(B∩C) = (A⨯B)∩(A⨯C)
(v) A⨯(B−C) = (A⨯B)−(A⨯C)
(vi) (A⨯B)∩(C⨯D) = (A∩C)⨯(B∩D)
(v) (A⨯B)∩(B⨯A) = (A∩B)⨯(B∩A)
(vi) If A⊂B and C⊂D, then A⨯C⊂B⨯D
Algebra of Real Functions
Addition
(f+g)(x)= f(x)+g(x)
Subtraction
(f-g)(x)= f(x)-g(x)
Multiplication by a scalar
(cf)(x) = c f(x)
Multiplication
(fg)(x)= f(x)g(x)
Quotient
$\left ( \frac{f}{g} \right )\left ( x \right )=\frac{f(x)}{g(x)}$
*To Solve Problems
For finding domain of function f(x)
(i) Find all the values of x for which the function is defined
(ii) Denominator should not be zero
(iii) Expression under the root should not be negative
For finding range of y=f(x)
(i) Find the domain for the function y=f(x)
(ii) Change the function y=f(x) as x=φ(y)
(iii) Solve x=φ(y)
(iv) Find the values of y for x in the domain of f. The values of y is the range.
For finding domain of function f(x)
(i) Find all the values of x for which the function is defined
(ii) Denominator should not be zero
(iii) Expression under the root should not be negative
For finding range of y=f(x)
(i) Find the domain for the function y=f(x)
(ii) Change the function y=f(x) as x=φ(y)
(iii) Solve x=φ(y)
(iv) Find the values of y for x in the domain of f. The values of y is the range.
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