Chapter 1: Sets

Chapter 1: Sets

• A set can be defined as a well-defined collection of objects
• An example of set: the collection of all rational numbers less than 10
• An example of non-set: the collection of all brilliant students in a class
• Sets are usually denoted by capital letters A, B, S,
• Elements of a set are usually denoted by small letters a, b, t, u
• If x is an element of a set S, then “x belongs to S” which is written as x ∈ S
• If y is not an element of a set S, then “y does not belong to S” which is written as y ∉ S

Methods for representing a set

• Roster or tabular form lists all the elements of a set
• Elements are separated by commas and enclosed within braces { }
• Order is immaterial
• Elements are not repeated
• Example: The set of letters forming the word ‘TEST' is {T, E, S}.
• Set-builder form means all the elements of a set possess a single common property
• No element outside the set possesses the property
• Example: The set {2, –2} can be written in the set-builder form as {x : x is an integer and x² - 4 = 0}.

Types of sets

• An empty set contains no element
• Empty sets are called a null set or a void set.
• Denoted by the symbol p or { }
• Example: The set {x : x ∈ N, x is an even number and 8 < x < 10} is an empty set
• A finite set is empty or consists of a definite number of elements
• An infinite set is not a finite
• Example: The set {x : x ∈ N and x is a square number} is an infinite set
• Example: The set {x : x ∈ N and x2 - 2x – 3 = 0} is a finite set as it is equal to {−1,3}
• Infinite sets cannot be described in the roster form
• The set of rational numbers cannot be described in this form
• Elements of this set do not follow any particular pattern

Equal and Unequal Sets

• Two sets A and B are equal if they have exactly the same elements
• Equal sets are written A = B
• Unequal sets are written A ≠ B
• Example: The sets A = {x: x ∈ N and (x-1)(x + 4) = 0} and B = {–4, 1} are equal sets
• Order of element does not matter
• Example: The sets A = {E, L, E, M, E, N, T, S} and B = {E, L, M, N, T, S} are equal since each element of A is in B, and vice-versa
• A set does not change if one or more elements of the set are repeated

Subsets

• Set A is a subset of set B if every element of A is also an element of B
• Written as A ⊂ B
• A ⊂ B if a ∈ A⇒a∈B
• A ⊂ B and B ⊂ A ⇔ A = B
• An empty set is a subset of every set
• Every set is a subset of itself
• If A ⊂ B and A ≠ B, then A is a proper subset of B
• B is called a superset of A

Singleton Set

• All sets with one element are singleton sets.
• Example: A = {-17} is a singleton set

Intervals

• Intervals are subsets of R
• Let a, b ∈ R and a < b
• {y : a <y<b} is an open interval and is denoted by (a, b)
• In the open interval (a, b), all the points between a and b belong to the open interval (a, b)
• a, b themselves do not belong to this interval
• {y : a ≤ y ≤ b} is a closed interval and is denoted by [a, b]
• In this interval, all the points between a and b as well as the points a and b are included
• [a, b) = {y : a ≤ y <b} is an open interval from a to b, including a, but excluding b
• (a, b] = {y : a < y ≤b} is an open interval from a to b, including b, but excluding a

Power set

• The collection of all subsets of a set A is called the power set of A
• Denoted by P(A)
• If A is a set with n(A) = m, then n[P(A)] = 2m

Venn diagrams

• Relationships between sets are represented by diagrams known as Venn diagrams
• A universal set is the super set of all sets under consideration
• Denoted by U

Union of sets

• The union of two sets A and B is the set which contains all those elements which are only in A, only in B and in both A and B
• Denoted by ‘A ∪ B’
• A∪ B = {x : x ∈ A or x ∈ B}

Properties of union of sets

• A ∪ B = B ∪ A (Commutative Law)
• (A∪B)∪C = A∪(B∪C) (Associative Law)
• A ∪ ø = A (ø is the identity of )
• A ∪ A = A (Idempotent Law)
• U ∪ A=U (Law of U)

Intersection of sets

• The intersection of two sets A and B is the set of all those elements which belong to both A and B
• Denoted by ‘A ∩ Β’
• A∩B={x:x∈ A and x ∈ B}

Properties of intersection of sets

• A ∩ B = B ∩ A (Commutative Law)
• (A∩B)∩C = A∩(B∩C) (Associative Law)
• ø ∩ A = ø (Law of ø)
• U ∩ A = A (Law of U)
• A ∩ A = A( Idempotent Law)
• A ∩(B∪C)=(A∩B)∪(A∩C) (Distributive law of ∩ on ∪)
• A ∪ (B∩C) = (A ∪ B)∩(A ∪ C)

Difference of sets

• The difference between the sets A and B (i.e., A – B, in this order) is the set of the elements which belong to A, but not to B
• A − B = {x: x ∈ A and x ∉ B}

Properties of operation of difference of sets

• A − B =A − (A ∩ B)
• For A ≠ B, A - B ≠ B - A
• For A = B, A − B = B − A = φ
• For B = φ, A – B = A and B − A = ø
• A – U = ø
• For A ⊂ B, A – B = ø, for this reason, A – U = ø
• The sets A – B, A ∩ B and B – A are mutually disjoint sets
• Any intersection of these two sets is a null set

Complement of a set

• If U is a universal set and A is a subset of U, then the complements of A are denoted by the set A'
• This is the set of all element of U which are not the elements of A
• A' = {x : x ∈U and x ∉ A} =U-A
• A' is also the subset of U

Properties of complement of a set

• (A')' = A
• A∪A'=U
• A∩ A' = ø
• φ' = U and U' = φ

De Morgan's laws

• For any sets A and B, (A∪B)' = A' ∩ B' (A∩B)' = A' ∪ B'

Finite Sets

• If A and B are finite sets, such that A ∩ B = 4, then n(A∪B) = n(A)+n(B)
• If A and B are finite sets, such that A∩B ≠ ¢, then, n(A∪B) = n(A)+n(B)-n(A∩B)
• If A, B and C are finite sets, then n(A∪B∪C)=n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C)

Chapter 2: Relations and Functions

• A Cartesian product of sets are constructed when provided with two non-empty sets P and Q
• The Cartesian product of P × Q is the set of all ordered pairs of elements from P and Q
• P × Q = {(p,q) : p∈ P and p∈Q}
• If either P or Q is a null set, then P × Q will also be a null set
• P × Q=¢
• In general, if A is any set, then A × ¢ = ¢
• Two ordered pairs are equal if and only if the corresponding first elements are equal and the second elements are also equal
• if (a, b) = (x, y), then a = x and b = y
• For any two sets A and B, A×B≠B×A
• If n(A) = p, n(B) = q, then n(A × B) = pq
• If A and B are non-empty sets and either A or B is an infinite set, then so is the case with A×B
• A×A×A = {(a,b,c): a,b,c ∈ A}
• Here, (a, b, c) is called an ordered triplet

Relation

• A relation R from a set A to a set B is a subset of the Cartesian product A × B
• Obtained by describing a relationship between the first element x and the second element y of the ordered pairs (x, y) in A × B
• The image of an element x under a relation R is y, where (x, y) ∈ R
• The set of all the 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 the second elements in a relation R from a set A to a set B is called the range of the relation R
• The whole set B is called the co-domain of the relation R
• 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, then n(A × B) = pq and the total number of relations is 2pq

Functions

• A relation f from a set A to a set B is a function if every element of set A has one and only one image in set B
• A function f is a relation from a non-empty set A to another non-empty set B, such that the domain of f is A and no two distinct ordered pairs in f have the same first element
• The function f from A to B is denoted by f : A→ B
• A is the domain and B is the co-domain of f
• If f is a function from A to B and (a, b) ∈ f, then f (a) = b, where b is called the image of a under f, and a is called the pre-image of b under f
• A function having either R (real numbers) or one of its subsets as its range is called a real-valued function
• If its domain is also either R or a subset of R, it is called a real function

Types of functions

• Identity function: The function f: R → R defined by y = f(x) = x, for each x ∈ R, is called the identity function
• R is the domain and range of f
• Constant function: The function f: R → R defined by y = f (x) = c, for each x ∈ R, where c is a constant, is a constant function
• The domain of f is R and its range is {c}
• Polynomial function: A function f: R → R is said to be a polynomial function if for each x ∈ R, y = f(x) = a₀ +a₁x+.......+aₙxⁿ, where n is a non-negative integer and a₀,a₁........aₙ∈R
• Rational function: The functions of the type f(x)/g(x), where f(x) and g(x) are polynomial functions of x defined in a domain and where g(x) ≠ 0, are called rational functions
• Modulus function: The function f: R → R defined by f(x) = |x|, for each x ∈ R, is called the modulus function
• f(x) = x, x ≥0
• f(x) = -x, x < 0
• Signum function: The function f: R → R defined by f(x) = 1, if x>0 f(x) = 0, if x = 0 f(x) = −1, if x < 0
• Has a domain of R and its range is the set {−1, 0, 1}
• Greatest Integer function: The function f: R → R defined by f(x) = [x], x ∈ R, assuming the value of the greatest integer less than or equal to x, is called the greatest integer function
• Linear function: The function f defined by f (x) = mx +c, for x ∈ R, where m and c are constants, is called the linear function
• R is the domain and range of f

Algebra of functions

• For functions f: X → R and g: X → R, we define (f+g): X → R by (f + g)(x) = f (x)+g(x), x ∈ X (f-g): X → R by (f-g)(x) = f (x)−g(x), x ∈ X (fg): X → R by (fg)(x) = f (x).g(x), x ∈ X (af): X → R by (af)(x)=af(x), x ∈ X and a is a real number g : X → R by g(x) = f(x) , x ∈ X and g(x) ≠ 0 f(x)/

Chapter 3: Trigonometric Functions

• Find the relationship between an angle and the length of a side

• Consider a circle of radius r having an arc of length l that subtends an angle ofradians, where l = re

Relations Between Radian and Degree Measure

• Radian measure = π/180 × Degree measure.
• one sixtieth of a degree is called a minute, written as 1'
• one sixtieth of a minute is called a second, written as1"
• 1° = 60' and 1' = 60"

Domain and range of trigonometric functions

Signs of Trigonometric Functions in Different Quadrants

Trigonometric identities

1 • cosec x = sin x 1 • sec x = COS X SinX • tan x = COS X 1 COS X • cot x = = tanx SIN X • Cos²x+Sin²X = 1 • 1+tan2 x= sec2 X • 1+CoT2 X = CoSEC 2x

Trigonometric ratios of allied angles

• sin (-x) = - sin x
• cos (-x) = cos x
• cos(π/2 -x)= sin x
• sin(π/2 - x) = cos x
• cos(π/2+x)= -sin x
• sin(π/2 + x)= cos x
• cos(π-x) = - cos x
• sin(π - x) = sin x
• cos(n+x)= - cos x
• sin(n+x) = -sinx
• cos(2n-x) = cos x
• sin(2n-x) = -sin x
• cos(2nn+x) = cosx, n ∈ Z
• sin(2n + x) = sinx, n ∈ Z

Sum and difference of two angles

• sin (x + y) = sin x cos y + cos x sin y • cos (x + y) = cos x cos y - sin x sin y • sin (x - y) = sin x cos y - cos x sin y • cos (x - y) = cos x cos y + sin x sin y -If none of the angles x, y and (x + y) is an odd multiple of π/2 then,

• tan(x + y) = tan x + tan y /1 - tan x tan y • cot(x + y) = cot x cot y - 1 / cot y + cot x -If none of the angles x, y and (x ± y) is a multiple of π, then • tan(x - y) = cot x cot y + 1/cot y - cot x

-Trigonometric ratios of multiple angles 1-tan²x/1+tan²x

• cos 2x = cos²x - sin²x = 2 cos²x - 1 = 1 - 2sin² x = 2tanx/1+tan²x
• sin 2x = 2 sin x cos x = 2 tanx / 1 - tan² x
• tan 2x =
• sin 3x = 3 sin x – 4 sin³ x
• cos 3x = 4 cos³x - 3 cos x -tan 3x = 3 tan x - tan³ x / 1 - 3 tan² x

Some more trigonometric identities

x + y/2 cosx - y/2

• sinx + siny = 2cos -cosx + cosy = -2sin x + y/2 sin - y/2 -sinx - siny = 2sin x + y/2 cosx - y/2
• cosx - cosy = 2sin x + y/2 sinx - y/2 cos + cos (x - y)
• 2 cos x cosy = cos (x + y) +
• 2 sin x sin y = cos (x + y) – cos (x - y)
• 2 sin x cos y = sin (x + y) + sin (x - y)
• 2 cos x sin y = sin (x + y) – sin (x - y)

General solutions of trigonometric equations

+nx = 0 ⇒ x = nπ, where ∈ Z -sin +cosx = 0 ⇒ x = (2n+1)π/2 where n ∈ Z +sin x = sin y ⇒ x = nπ+(−1)ⁿy, where ,n ∈ Z +cos x = cos y ⇒ x = 2nn #+y, where n ∈ Z +tan x = tan y ⇒ x = nπ + y, where n ∈ Z

Chapter 4: Principle of Mathematical Induction

• There are some mathematical statements or results that are formulated in terms of n, where n is a positive integer.
• To prove such statements, the well-suited principle that is used, based on the specific technique, is known as the principle of mathematical induction.
• To prove a given statement in terms of n, we assume the statement to be P(n).
• Thereafter, we examine the correctness of the statement for n = 1 i.e., for P(1) to be true.
• Then, assuming that the statement is true for n = k, where k is a positive integer, we prove that the statement is true for n = k + 1 i.e., the truth of P(k) implies the truth of P(k + 1).
• Then, we say that P(n) is true for all natural numbers n.

Chapter 5: Complex Numbers and Quadratic Equations

• A number of the form a + ib, where a and b are real numbers and i = √–1, is defined as a complex number.
• For the complex numbers z = a + ib, a is called the real part (denoted by Re z) and b is called the imaginary part (denoted by Im z) of the complex number z.
• Two complex numbers z₁ = a + ib and z₂ = c + id are equal if a = c and b = d.

Addition of complex numbers

• Let z₁ = a + ib and z₂ = c + id be any two complex numbers. The sum is defined as z₁ + z₂ = (a + c) + i (b + d).

Properties of addition of complex numbers

• (i) Closure law: Sum of two complex numbers is also a complex number.
• (ii) Commutative law: For two complex numbers z₁ and z₂, z₁ + z₂ = z₂ + z₁
• (iii) Associative law: For any three complex numbers z₁, z₂ and z₃, (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)
• (iv) Existence of additive identity: There exists a complex number 0 + i0 (denoted by 0), called the additive identity or zero complex number, such that for every complex number z, z + 0 = z
• (v) Existence of additive inverse: For every complex number z = a + ib, there exists a complex number –a + i(−b) [denoted by -z], called the additive inverse or negative of z, such that z + (−z) = 0
• Given any two complex numbers z₁ and z₂, the difference z₁ – z₂ is defined as z₁ - z₂ = z₁ + (-z₂)

Multiplication of complex numbers

• For two complex numbers z₁ and z₂, such that z₁ = a + ib and z₂ = c + id, the multiplication is defined as z₁ z₂ = (ac – bd) + i(ad + bc).

Properties of multiplication of complex numbers

• Closure law The product of two complex numbers is also a complex number.
• Commutative law For any two complex numbers z₁ and z₂, z₁z₂ = z₂z₁.
• Associative law For any three complex numbers z₁, z₂ and z₃, (z₁z₂) z₃ = z₁ (z₂z₃)
• Existence of multiplicative identity There exist a complex number 1 + i 0 (denoted as 1), called the multiplicative identity, such that for every complex numbers z, z.1 = z

Existence of multiplicative inverse: For every non-zero complex number z a + ib (a 0,b 0), we have the complex number a+-ba²+b²√ a²+b²= a²+b² + i( a²+b² ) a+b2 (denoted by 1/z or z ⁻¹), called the multiplicative inverse of z, such that z. 1/z =1 Distributive law: For any three complex numbers z₁, z₂ and z₃,z₁ (z₂ + z₃) = z₁z₂ + z₁z₃ (z₁ +z₂) z₃ = z₁z₃+ z₂z₃ Given any two complex number z₁ and z₂, where z₂ 0, the quotient z₁/ z₂ is defined as

• z₁/ z₂ * =0 (1/ z₂)

Powers of i

For any integer k, i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = −1, i⁴ᵏ⁺³ = −i

If a and b are negative real numbers, then √a × √b √ab.

• Identities for complex numbers* For two complex numbers z₁ and z₂.
• (z₁ + z₂)² = z₁² +2z₁z₂ + z₂²
• (z₁- z₂)² = z₁² -2z₁z₂ + z₂²
• z₁² - z₂² = (z₁+ z₂)( z₁ - z₂)
• (z₁ + z₂)^3 = z₁^3 +3z₂z₂ +3z₁z₂²+ z₂^3 (z₁- z₂)^3 = z₁^3 -3z₂z₂ +3z₁z₂² -z₂^3

Modulus and conjugate of complex numbers

• The modulus of a complex number z = a + ib, is denoted by/z/, and is defined as the nonnegative real number √a² + b², i.e., √a²+b².
• The conjugate of a complex number z = a + ib, is denoted by Z, and is defined as the complex number a ib, i.e., z =a-ib.

Properties of modulus and conjugate of complex numbers

For any three complex numbers z, z₁, z₂,

• z₁ z = /z/² or z.z =z/=2 1
• /z. z ² /-/z1/ . /z2/ /, provided /z₂/0 z
• z*₁_ =/z₁/= √₂ √₂
• √z₁z₂ ̅ =√z₁/√z₂ √z₁=/zī

The polar form of the complex number z = x + iy, is √ x² + y² (modulus of z) and cos0 = x/_*/2, i.e., √. .sin √ z= x-iy = re0+√ π

Is such that - <0 < n is called the principle arguments z.

Solution for the given equation

a≠ 0 is called solutions of quadratic equations by x="b-b+4ac/2e A polynomial equation of degree n has a root.

Chapter 6: Linear and in equalities.

• In linear qualities only x10 qualities in 2 variable.

• Any set of value of x which it a member of solution is of lineal of qualities,

• Solving qualities Algebricolys: + Equal number added to side of qualities without effecting sin. + When both sides and multiply by - value, is side reverse.

  +  Graphical representation:-
  

• Represent x < a on number line, incide number dark line to the side from.

• Representing x 1 <a, dark number line to the side from.

graphical solution of linear qualities and 2 variable

• Solution solution reason system and system.
• Identify to have plant equation.

• (1A) < O are green x = 2ª - a √ac-b ² /a . is called form of equation

Chapter 7: Permutations and combination

• Events found fundamental princible = m x h/ is m different way found h.
• Arrg.definite = permutations/

Notation

• Factorial notation represented product notation numbers L/E - Nx/x/(1)/0

Number of permutation

• Repetition allowed
• Repetition - = p

Distinct permulations

• P elements kind of distinct n elements = n
  • N Ob. - distinct types = equation A distinct selection is = arrangement- A number = combination

Combination, Equation = ⁰ C

propert

• Cr

ch8 binomial thereal

• Pascal triangle
• Expansion = N c a n * 150

Ch-9 sequence and series

A sequence : arrangements numbers = to number of some role, and function

• Series 1/2/3/ equation = 123/12/3/ is to be Geometric progressions = Same throughout = to be .

• Series progression same throught formula/arithmetic progressions.

Chapter 10 and series

• Slope of line grade.

tan = -1, slope * 13 - 0* line and y intercept form/2 parts intercept form

ch-1 conic

• Axis equation a normal distance from angle which
• A point equation general
• Conic equation

double point double nap

(Axis line) fixed line called vortex.

Chapter 14, Limits and derivatives

• It's the function for which exists values for x to near left of a.
• Right and same thing is right in.

Algebra limits

Algebra derivatives

= h (x) #0 -h(π).

Standard Derivative [ 1 *] Derivative

Chapter 14; The denial of statement, statement, or, to show the statements.

Chapter 15

• Med, modes idea central points- to - Dispersion.

• Max value - min value Range quartile Deviation.

Deviation to mean + standard of equation

Chapter 16

• If 2 coins at lead/18 - The probability outcome of formula equation. , Is the following are, the sides, given is side from.

• A is and - A and B. a ∩ to.