Elements of Set theory
Indices and Logarithms Commercial Arithmetic
Differential Calculus Determinants
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Elements of Set theory
Indices and Logarithms Commercial Arithmetic
Differential Calculus Determinants
Click to exit
Understanding set theory helps people to … •see
things in terms of systems
•organize •begin
things into groups
to understand logic
Set: A well-defined collection of distinct
objects is called a set. Element: An element is an object contained
in a set. Notation of Sets: Capital letters are usually
used to denote or represent a set.
Representation of Sets: There are two
methods of representing a set. (i) Roster Method (ii) Set builder form. Finite and Infinite Sets: A set is finite if it
contains a specific number of elements. Otherwise, a set is an infinite set.
Null Set or Empty Set or Void Set: A set
with no elements is an empty set.
Singleton Set or Singlets: A set consisting
of a single element is called a singleton set or singlet. The cardinality of the singleton set is 1. Equivalent Sets: Two finite sets A and B are
said to be equivalent sets if cardinality of both sets are equal i.e. n (A) = n (B).
Equal Sets: Two sets A and B are said to be
equal if and only if they contain the same elements i.e. if every element of A is in B and every element of B is in A. We denote the equality by A = B. Cardinality of a Set A: The number of
elements in a finite set A, is the cardinality of A and is denoted by n(A).
Universal Set: In any application of the
theory of sets, the members of all sets under consideration usually belong to some fixed large set called the universal set. Power Set: The family of all subsets of any
set S is called the power set of S. We denote the power set of S by P (S).
Write all definitions.
Upper case designates set name Lower case designates set elements { } enclose elements in set is (or is not) an element of ∈ or ⊆ is a subset of (includes equal sets) ⊂ is a proper subset of ⊄ is not a subset of ⊃ is a superset of | or : such that (if a condition is true) | | the cardinality of a set
S={a, b, c} refers to the set whose elements are a, b and c. a∈S means “a is an element of set S”. d∉S means “d is not an element of set S”.
{x ∈S | P(x)} is the set of all those x from S such that P(x) is true. E.g., T={x ∈Z | 0
{b,a,c}, {c,b,a,b,b,c} all represent the same set. 2.Sets can themselves be elements of other sets, e.g., S={ {cat, rat}, {Pot, Pan}, …}
Roster method: The method of specifying a
set consists of surrounding the collection of elements with braces.
Set Builder method: This has the general form
{variable | descriptive statement }. The vertical bar (in set builder notation) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate
Subset: If every element of Set A is also contained in Set B, then Set A is a subset of Set B
A is a proper subset of B if B has more elements than A. •
Intersection: The intersection of two sets A
and B is the set containing those elements which are elements of A Written as A B
and elements of B.
Union: The union of two sets A and B is the
set containing those elements which are elements of A
or elements of B.
Written as A ∪ B
Commutative operations: A B=B A A∩B=B∩A Distributive Law: A A
∩
(B (B
∩
C )= (A C )= (A
∩
B) B)
∩
(A (A
∩
C) C)
Associative Law: A A
∩
( B ∩ C )= (A ∩ B) ∩ C (B C )= (A B) C
Other Properties: A =A A∩ = A A=A A∩A=A
Write proofs for • • •
A A A
∩
B=B∩A ( B ∩ C )= (A (B C )= (A
B) ∩ (A B) C
C)
Subset: If every element of Set A is also contained in Set B, then Set A is a subset of Set B
A is a proper subset of B if B has more elements than A. •
Let U be a universal set. The complement of a set A is defined to be the set of all elements which are in U and not in A. The complement of A is denoted by A’ or A or Ac. (i.e.) A’ = {x | x∈U, x∉A}
The difference of sets A and B is defined to be the set which contains all those elements in A which are not in B. The difference of set A and B is denoted by A – B (i.e.) A-B = {x | x∈A, x∉B} Similarly B-A = {x | x∈B, x∉A} Note:
A-B ≠ B-A
Suppose A and B are sets. Then A is called a subset of B: A
B
iff every element of A is also an element of B. Symbolically, A ⊆ B ⇔ ∀x, if x∈A then x ∈B. A ⊄ B ⇔ ∃ x such that x∈A and x∉B.
Venn Diagram:
A Venn diagram is a pictorial representation of sets by set of points in the plane. The universal set U is represented pictorially by interior of a rectangle and the other sets are represented by closed figures viz circles or ellipses or small rectangles or some curved figures lying within the rectangle.
Venn diagrams show relationships between sets and their elements Sets A & B
U A
B
Universal Set
A is a subset of B and is represented as shown in the venn diagram.
(A∪B)’ = A’ ∩B’ (A∩B)’ = A’ ∪B’
Prove (A∩B)’ = A’ B’ using Venn diagram.
Laws of Indices
If m and n are positive integers, and a≠0 then Product Law: am.an = am + n m n Quotient Law: a /a = am - n Power Law: (am)n = amn
(ab)m = am . bm am/n=(am) 1/n=n√am ao = 1 a-1=1/a
1.
If x = yz, y = zx, z = xy show that xyz=1
2.
Show that
is independent of a and n.
Definition: The logarithm of a positive
number to a given base is the index of the power to which the base is raised to equal the number. Consider N = a N is a positive number, is the base and is the index of the power. From the definition we can write the equation N = a as x
x
Product Rule:
Quotient Rule:
Power Rule:
(
Rule of Change of base:
1.
Find the value of
2.
Prove that
Common Logarithms:
Logarithms with base 10 are called common logarithms Natural Logarithms:
Logarithms with base e are called Napier or Natural logarithms. The value of e lies between 2 and 3.
Characteristics and Mantissa:
The logarithm of a number contains two parts 1) the integral part and 2) the decimal part. The integral part is called char characteristics acteristics and the decimal part is called the t he mantissa
For a number greater than one the characteristicc is the number, characteristi number, one less less than than the number of digits before the decimal point and is positive.
For a number less than one the characteristic char acteristic is the number one more than the number of zeros between the decimal point and the first non-zero digit in the number expressed as a decimal and it is negative.
1.
Find the number of digits in 715
2.
When 2-5*3-4 is expressed as a decimal find the position of the first significant figure.
Interest is the cost of borrowing money. An interest rate is the cost stated as a percent of the amount borrowed per period of time, usually one year.
Simple interest is calculated on the original principal only.
Compound interest is calculated each period on the original principal and all interest accumulated during past periods. Although the interest may be stated as a yearly rate, the compounding periods can be yearly, semi annually, quarterly, or even continuously.
When the interest is compounded halfyearly or quarterly the interest which becomes due at the end of the year on a loan is called the Effective Rate of Interest.
The given rate for calculating interest for half- yearly or quarterly is called the Nominal Rate.
When Compound interest is considered.
Presen When Simple interest is considered.
The simple interest on a certain principal for 5 years is Rs.360 and the interest is 9/25 of the principal. Find the principal and the interest rate. 2. The population of a country increases every year by 2.4 percent of the population at the beginning of the year. In what time will the population double itself? Answer to the nearest year. 3. Compute the interest on Rs.1000 for 10 years at 4% per annum, the interest being paid annually. 1.
To honour any transaction it is necessary to have documents like bill of exchange or promissory note according to which the debtor has to pay to the creditor a specified sum of money on a specified date.
Two kinds of bills 1.Bill of exchange after date 2.Bill of exchange after sight
Face value is the value of a coin, stamp or paper money, as printed on the coin, stamp or bill itself by the minting authority. While the face value usually refers to the true value of the coin, stamp or bill.
True Discount: The interest on the present value is called True
Discount
T.D = A – P or T.D = Pni
Where A – Total Amount P – Principal Banker’s Discount: The interest on sum due is called Banker’s
Discount.
B.D = Ani
Where A – Amount n – number of years i- interest percent
Banker’s Gain = B.D – T.D
Or
Banker’s gain is interest on true discount
The date on which the bill of exchange becomes due is called the nominal due date or simply due date. There is a legal custom to allow 3 days of grace from the due date to encash a bill. If these days are added to the due date, we get the date on which the bill becomes legally due.
Find the date of drawing the bill. Find the date of maturity. Find the date on which the bill was discounted. Count the number of days from the date of discount to the date of maturity. Add three days grace period to this to give the legal due date.
A bill of Rs.3,225 was drawn on 3 rd February 1965 at 6 months due date and discounted on 13th March 1965 at the rate of 8% per annum. For what sum was the bill discounted and how much did the banker gain in this? 2. The difference between true and bankers discounts on a certain bill due in 4 months is 50 paise. If the rate of interest is 6 percent, find the amount of the bill. 1.
Differential calculus can be considered as mathematics of motion, growth and change where there is a motion, growth, change. Whenever there is variable forces producing acceleration, differential calculus is the right mathematics to apply. Application of derivatives are used to represent and interpret the rate at which quantities change with respect to another variable. Differential equations are powerful tools for modeling data.
A differential equation is an equation involving derivatives of an unknown function and possibly the function itself as well as the independent variable. The order of a differential equation is the highest order of the derivatives of the unknown function appearing in the equation
If y is a function of x, that is y = f(x), we write its gradient function as .
if y = xn, then If y=c, then Where c is a constant
It is appropriate to use this rule when you want to differentiate two functions which are multiplied together. If u and v are functions of x, then
If u and v are functions of x, then In the rule which follows we let u stand for the function in the numerator and v stand for the function in the denominator.
Let y be a function of u and u be a function of x. Then
Let y be a function of u, u is a function of v and v is a function of x. Then
Differentiate the following with respect to x. • • •
The diagram below shows part of a function y = f(x).
The point A is a local maximum and the point B is a local minimum. At each of these points the tangent to the curve is parallel to the x-axis so the derivative of the function is zero. Both of these points are therefore stationary points of the function. The term local is used since these points are the maximum and minimum in this particular region. There may be others outside this region.
The rate of change of a function is measured by its derivative. When the derivative is positive, the function is increasing, when the derivative is negative, the function is decreasing. Thus the rate of change of the gradient is measured by its derivative, which is the second derivative of the original function. In mathematical notation this is as follows.
At the point (a, b) If and
then point(a,b)
is a local maximum. If
and
Local minimum.
then point(a,b) is a
There are two general rules:
න නሾሺ ሻ
Few other rules:
Other rules:
Other standard results:
න(න+
Find the minimum average cost for the average cost function and show that at the minimum average cost, marginal cost and average cost are equal. • Integrate the following with respect to x. 1.3x3+7x2-2x+1 2.(1-x)3 •
3.
Definition of a Matrix: A
rectangular array of entries is called a Matrix. The entries may be real, complex or functions. The entries are also called as the elements of the matrix. The rectangular array of entries are enclosed in an ordinary bracket or in square bracket. Matrices are denoted by capital letters.
Let A = [aij] be a square matrix. A determinant formed by the same array of elements of the square matrix A is called the determinant of the square matrix A and is denoted by the symbol det.A or | A|. The determinant of a square matrix will be a scalar quantity. i.e., with a determinant we associate a definite value, whereas a matrix is essentially an arrangement of numbers and has no value.
If the rows and columns of a determinant are inter-changed, the value remains unaltered. If any two rows (columns) of a determinant are identical, its value of the determinant is zero. If any two rows (columns) of a determinant are interchanged, the value of the determinant is (-1) times the original determinant.
If all the elements of one row (column) of a determinant is multiplied by k, the value of the new determinant is k times the original determinant. If to any row or column of a determinant, a multiple of another row or column is added, the value of the determinant remains the same. If some or all the elements of a row (or column) of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two or more determinants.
The sum of the products of the elements of any row (column) with their corresponding cofactors is equal to the value of the determinant. The sum of the products of the elements of any row (column) and the cofactors of the corresponding elements of any other row (column) is zero.
• • • • • •
Row Matrix Column Matrix Square Matrix Diagonal Matrix Scalar Matrix Identity or Unit Matrix
• • • • • •
Null Matrix or Zero matrix Upper triangular matrix Lower triangular matrix Transpose of a matrix Symmetric matrix Skew-symmetric matrix
Write all different types of Matrix
Only matrices of the same order can be added or subtracted. The result from the sum or difference is then determined by adding or subtracting the corresponding elements.
7 3
9 6
42 + 3 13
5
7 + 42 = 3 + 13 49 = 16
43 22
+ 34 6 + 16
9
21
26
34 16
18 21
+ 21 3 + 18
5
Two matrices A and B are conformable to matrix multiplication only if the number of columns in A equals the number of rows in B. AB has the same number of rows as A and the same number of columns as B. The ijth element of AB is the inner product of the ith row of A and jth column of B.
a a a
11
21
31
a12 a22 a32
b b
11
b12
b13
21
b22
b23
a × b + a × b = a ×b + a ×b a ×b + a ×b
11
11
12
21
a11
21
11
22
21
a21
31
11
32
21
a31
×b + a ×b ×b + a ×b ×b + a ×b 12
12
22
a11
12
22
22
a21
12
32
22
a31
×b + a ×b ×b + a ×b ×b + a ×b 13
12
23
13
22
23
13
32
23
If A = B= Find 1) A+B 2) A-B
If A = B= Find 1) AB 2) BA
The inverse of a square matrix A is a matrix whose product with A is the identity matrix I. The inverse matrix is denoted by A -1 .
The concept of “dividing” by A in matrix algebra is replaced by the concept of multiplying by the inverse matrix A-1. An inverse matrix A-1 has the following properties: A-1 A = A A-1 = I A-1 unique for given A.
The (+1) and (–1) factors in the expansion are decided on according to the following rule: ij If A is written in the form A = (a ), the product of aij and its minor in the expansion of determinant | A| is multiplied by (–1)i+j.
Therefore, because the element 1 in the example is the element a11, its product with its minor is multiplied by (–1)1+1 = +1. For element 2, which is a12 , its product with its minor is multiplied by (–1)1+2= –1
The determinant of a square matrix of order n, (that is, An×n = (aij); i, j = 1, 2, …, n) is the sum of all possible products of n elements of A such that each product has one and only one element from every row and column of A, the sign
of a product being (–1)i+j.
The determinant of a square matrix A, denoted by |A|, is a polynomial of the elements of a square matrix. It is a scalar. It is the sum of certain products of the elements of the matrix from which it is derived, each product being multiplied by +1 or –1 according to certain rules
Consider the linear equation a1x+b1y+c1z = d1 a2x+b2y+c2z = d2 a3x+b3y+c3z = d3 Denote A=
X= E= Then AX = E X = A-1 E
The rank of a matrix A is the order of its largest non-zero minor. Minor: The determinant of a submatrix of
order r of a given matrix will be called a minor of order r of the matrix.
Operations that can be performed without altering the solution of a linear system. 1.Change any two rows. 2.Multiply every element in a row by a non-zero constant. 3.Add elements of one row to corresponding elements of another row.
Consider the linear equation a1x+b1y+c1z = d1 a2x+b2y+c2z = d2 a3x+b3y+c3z = d3 Denote A =
and B=
The augmented matrix AB = Find the rank of matrix A
1. 2.
3. 4.
Find the rank of augmented matrix AB using row transformation. Using row transformation, transform the augmented matrix AB to an Echelon matrix. Write the system of equations corresponding to the resulting matrix. Use back-substitution to find the system’s solution