Sets/Relation/Function

1.1 Introduction

• The theory of sets was developed by German mathematician Georg Cantor.
• Sets are used to define the concepts of relations and functions.
• The study of geometry, sequences, probability, etc. requires the knowledge of sets.

1.2 Sets And Their Representations

• A set is a well-defined collection of objects.
• Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
• The elements of a set are represented by small letters a, b, c, x, y, z, etc.
1.2.1 Representation Of Sets
• There are two methods of representing a set:
• Roster or tabular form
• Set – builder form.
1.2.1.1 Roster Form:
• In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.
• Example: The set of all even positive integers less than 7 is described in roster form as { 2, 4, 6 }.
1.2.1.2 Set-Builder Form
• In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
• Example: In the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property.
• Denoting this set by K, we write,

K = { x : x is a vowel in English alphabet }

Example:
• Write the set L = {1, 4, 9, 16, 25, . . . }in set-builder form.
Solution:
• We may write the set L as

L = {x : x is the square of a natural number}

• Alternatively, we can write

L = {x : x = n2 , where n  N}

1.3 The Empty Set

• A set which does not contain any element is called the empty set or the null set or the void set.
• The empty set is denoted by the symbol  or { }.
Example:
• D = {x : x2 = 4, x is odd}. Then D is the empty set because the equation x2 = 4 is not satisfied by any odd value of x.

1.4 Finite and Infinite Sets

Finite Sets
• A set which is empty or consists of a definite number of elements is called finite.
Example:
• Let W be the set of the days of the week. Then W is finite.
Infinite Sets
• An infinite set is a set with an infinite number of elements.
Example:
• The set of all integers, {….., -1, 0, 1, 2, …..}

1.5 Equal Sets

• Two sets A and B are said to be equal if they have exactly the same elements and we write A = B.
Example:
• Let A be the set of prime numbers less than 6 and P the set of prime factors of 30.
• Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6.

1.6 Subsets

• A set P is said to be a subset of a set Q if every element of P is also an element of Q.
• It is expressed as P  Q;The symbol  stands for ‘is a subset of’.
Example:
• Let A = { a, e, i, o, u } and B = { a, b, c, d }. Is A a subset of B?
Solution:
• No, A is not a subset of B as every element of A is not an element of B.

1.7 Power Set

• The collection of all subsets of a set A is called the power set of A.
•  It is denoted by P(A).
Example:
• If A = { 1, 2 }, then P(A) = { , {1}, {2}, {1, 2} }
Note:

Cardinal Number :- The total no. of elements are known as Cardinal no.

• If A is a set with n(A) = m, then it can be shown that n[P(A)] = 2m .

1.8 Universal Set

• A Universal Set is the set of all elements under consideration, denoted by capital U.
• All other sets are subsets of the universal set.
Example:
• Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets.
• A = {x : x is a factor of 60}
• B = {x : x is a prime number}
Solution:
• The elements of sets A and B can only be selected from the given universal set U.
• A = {5, 6, 10, 12}
• B = {5, 7, 11}

1.9 Venn Diagrams

• A Venn diagram is a drawing in which circular areas represent groups of items which share a common property.
• The drawing consists of two or more circles, each representing a specific group or set.

Example:
• Given U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 5, 6 } and B = { 3, 9 }, draw a Venn diagram to represent these sets.
Solution:
• Given
• U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 },
• A = { 1, 2, 5, 6 } and
• B = { 3, 9 }.

1.10 Operations on Sets

• There are some operations which when performed on two sets give rise to another set. They are:
• Union of sets
• Intersection of sets
• Difference of sets
• Complement of a set
1.10.1 Union of Sets:
• The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both).
• The symbol ‘’ is used to denote the union.
• Symbolically, we write A  B = {x : x  A or x  B}
• The union of two sets can be represented by a Venn diagram as shown in figure.

Example:
• Let A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Find A  B?
Solution:
• We have A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Then A  B = { 2, 4, 6, 8, 10, 12 }
Properties of the Operation of Union:
• Commutative law : A  B = B  A
• Associative law :  (A  B) C = A  (B C)
• Law of identity element : A   = A
• Idempotent law : A  A = A
• Law of U : U  A = U

1.10.2 Intersection of Sets:
• The intersection of sets A and B is the set of all elements which are common to both A and B.
• The symbol  is used to denote the intersection.
• Symbolically we write it as A  B = {x : x  A and x  B}
• The shaded portion in the figure indicates the intersection of A and B.

Properties of Operation of Intersection:
• Commutative law: A  B = B  A
• Associative law: (A  B) C = A  (B  C)
• Law of  and U:   =
• Idempotent law: A  A = A
• Distributive law:
1.10.3 difference of Sets:-
• The difference of the sets A and B in this order is the set of elements which belong to A but not to B.
• Symbolically, we write A – B and read as “A minus B”.
Example:
• Let A = { 1, 2, 3, 4, 5, 6 }, B = { 2, 4, 6, 8 }. Find A – B and B – A.
Solution:
• A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but not to B
• B – A = { 8 }, since the element 8 belongs to B and not to A.

1.11 Complement of a Set

• Let U be the universal set and A is a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A.
• Symbolically, we write A’ to denote the complement of A with respect to U. Thus,

A’ = {x : x  U and x  A}

Example:
• Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } and A = { 1, 3, 5, 7, 9 }. Find A’.
Solution:
• 2, 4, 6, 8, 10 are the only elements of U which do not belong to A.
• Hence A’ = { 2, 4, 6, 8, 10 }
Properties of Complement Sets:
• Complement laws:
• A  A’ = U
• A  A’ =
• De Morgan’s law:
• (A  B)’ = A’  B’
• (A  B)’ = A’  B’
• Law of double complementation:
• (A’)’ = A
• Law of empty set and universal set
Formulae:

n(A  B ) = n(A) +  n(B)  –  n(A  B)

n(A  B  C) = n(A) + n(B) + n(C) – n(A  B) – n(B  C) – n(A  C) + (A  B  C)

Example:
• In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach both physics and mathematics. How many teach physics?
Solution:
• Let M denote the set of teachers who teach mathematics and P denote the set of teachers who teach physics.

n (M  P) = 20,

n(M) = 12

n(M  P) = 4

• To find n(P)

n (M  P) = n(M) + n(P) – n(M  P)

20 = 12 + n(P) – 4

n(P) =12

Thank You:- Prof Rajeev Tripathi

• Please email me at stephenhawking1982@gmail.com  and help me to improve this

Limit and Continuity

In mathematics, a limit is the value that a function or sequence “approaches” as the input or index approaches some value.[1] Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.

In formulas, a limit is usually written as

$\lim_{n \to c}f(n) = L$

and is read as “the limit of f of n as n approaches c equals L“. Here “lim” indicates limit, and the fact that function f(n) approaches the limit L as n approaches c is represented by the right arrow (→), as in

$f(n) \to L \ .$
For example, if

$f(x) = \frac{x^2 - 1}{x - 1}$

then f(1) is not defined (see division by zero), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2

In other words, $\lim_{x \to 1} \frac{x^2-1}{x-1} = 2$

This can also be calculated algebraically, as $\frac{x^2-1}{x-1} = \frac{(x+1)(x-1)}{x-1} = x+1$ for all real numbers x ≠ 1.

Now since x + 1 is continuous in x at 1, we can now plug in 1 for x, thus $\lim_{x \to 1} \frac{x^2-1}{x-1} = 1+1 = 2$

That means if you want to solve a Limit based question, you have to Factorise given function, then put the value.

Calculas

Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus, and integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that elementary algebra alone cannot

Arithmatics

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory

Totally dependent on the operation of Addition,Subtraction, Multiplication and Division.

Trigonometry/Function

If one of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

• Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

$\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.$

• Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
$\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.$
• Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
$\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{a}{\,c\,}*\frac{c}{\,b\,}=\frac{a}{\,c\,} / \frac{b}{\,c\,}=\frac{\sin A}{\cos A}\,.$

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).

The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:

$\csc A=\frac{1}{\sin A}=\frac{\textrm{hypotenuse}}{\textrm{opposite}}=\frac{c}{a} ,$
$\sec A=\frac{1}{\cos A}=\frac{\textrm{hypotenuse}}{\textrm{adjacent}}=\frac{c}{b} ,$
$\cot A=\frac{1}{\tan A}=\frac{\textrm{adjacent}}{\textrm{opposite}}=\frac{\cos A}{\sin A}=\frac{b}{a} .$
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