**Set:**

**A set is a well-defined collection of objects or elements. Each element in a set is unique.**

**Usually but not necessarily a set is denoted by a capital letter e.g. A, B, … U, V etc. and the elements are enclosed between brackets { }, denoted by small letters a , b , ….x , y etc.**

**For example:**

**A = Set of all small English alphabets**

**= {a , b , c ,… , x , y , z}**

**B = Set of all positive integers less than or equal to 10**

**= {1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10}**

**R = Set of real numbers = {x: − ∞ < x < ∞}**

**The elements of a set can be discrete (e.g. set of all English alphabets) or continuous (e.g. set of real numbers).**

**The set may contain finite or infinite number of elements. A set may contain no elements and such a set is called Void set or Null set or empty set and is denoted by φ (phi).**

**The number of elements of a set A is denoted as n(A) and hence n( φ) = 0 as it contains no element.**

**Union of Sets:**

**Union of Sets:**

**Union of two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is ‘ ∪ ‘**

**i.e. A∪B = Union of set A and set B**

**= {x: x ∈ A or x∈B (or both)}**

**Example:**

**If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A∪B∪C = {1, 2, 3, 4, 5, 6, 8}**

**Intersection of Sets:**

**Intersection of Sets:**

**It is the set of all the elements, which are common to all the sets.**

**The symbol used for**

**intersection of sets is ‘ ∩ ‘.**

**i.e. A∩B = {x: x ∈ A and x∈ B}**

**Example:**

**If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A ∩ B ∩ C = { 2 }**

**Remember that n(A ∪ B) = n(A) + n(B) − n(A ∩ B)**

**Difference of two sets:**

**Difference of two sets:**

**The difference of set A to B denoted, as A – B is the set of those elements that are in the set A but not in the set B**

**i.e. A – B = {x: x ∈ A and x ∉ B}.**

**Similarly B − A = {x: x ∈ B and x ∉ A}.**

**In general A − B ≠ B − A.**

**Example:**

**If A = {a, b, c, d} and B = {b , c , e , f}**

**then A − B = {a, d} and B − A = {e , f}**

**Subset of a Set:**

**Subset of a Set:**

**A set A is said to be a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘ ⊆ ‘**

**i.e. A ⊆ B <=> (x ∈ A**

**=> x ∈ B)**

**Each set is a subset of its own set. Also a void set is a subset of any set.**

**If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A ⊂ B.**

**If set B has n elements then total number of subsets of set B is 2 ^{n}**

**Example:**

**If A = {a, b, c, d} and B = {b, c, d} then B ⊂ A or equivalently A ⊃ B (i.e A is a super set of B).**

**Equality of Two Sets:**

**Equality of Two Sets:**

**Sets A and B are said to be equal if A ⊆ B and B ⊆ A and we write A = B.**

**Universal Set:**

**Universal Set:**

**As the name implies, it is a set with collection of all the elements and is usually denoted by U.**

**e.g. set of real numbers R is a universal set whereas a set A ={x: x ≤ 3} is not a universal set as it does not contain the set of real numbers x >3.**

**Once the universal set is known, one can define the Complementary set of a set as the set of all the elements of the universal set which do not belong to that set.**

**e.g. If A = {x: x ≤ 3} then A˜ = complimentary set of A = {x: x > 3}.**

**Hence we can say that A ∪ A˜ = U i.e. Union of a set and its complimentary is always the Universal set and A ∩ A˜ = φ**

**i.e intersection of the set and its complimentary is always a void set.**

**Cartesian product of Sets:**

**Cartesian product of Sets:**

**The Cartesian product (also known as the cross product) of two sets A and B , denoted by A x B (in the same order) is the set of all ordered pairs (x , y) such that x ∈ A and y ∈ B.**

**What we mean by ordered pair is that the pair (a, b) is not the same pair as (b, a) unless a = b. It implies that**

**A × B ≠ B × A in general. Also if A contains m elements and B contains n elements then A x B contains m x n elements.**

**Similarly we can define A × A = {(x, y); x ∈ A and y ∈ A}. We can also define cartesian product of more than two sets.**

**e.g. A _{1} × A_{2} × A_{3} . . . . × A_{n} = {(a_{1}, a_{2}, . . . , a_{n}) : a_{1} ∈ A_{1} , a_{2} ∈ A_{2}, . . . , a_{n} ∈ A_{n}}**

**Example: **

**Sixty five percent of children in a sport club play football , 70 percent play volley-ball and 75 per-cent play basket ball. What is the smallest percentage of children playing all the three games?**

**Solution: Out of 100 children,**

**Number of children who do not play football = 100 − 65 = 35**

**Number of children who do not play volley-ball = 100 − 70 = 30**

**Number of children who do not play basket ball = 100 − 75 = 25.**

**So, the maximum number of children who do not play atleast one game**

**= 35 + 30 + 25 = 90.**

**Thus the minimum number of children who play all the three games = 100 − 90 = 10.**

**Hence the smallest percentage of children playing all the three games = 10%.**

**Note: The greatest percentage of children playing all the three games = min(65%, 70%, 75%} = 65%.**

**NUMBER THEORY:**

**NUMBER THEORY:**

**Natural Numbers:**

**Natural Numbers:**

**The numbers 1 , 2 , 3 , 4 … are called natural numbers, their set is denoted by N. Thus N = {1 , 2 , 3 , 4 , 5 …}**

**Integers:**

**Integers:**

**The numbers … −3 , −2 , −1 , 0 , 1 , 2 , 3 … are called integers and the set is denoted by I or Z.**

**Thus I (or Z) = {… −3 , −2 , −1, 0,1, 2, 3 …}.**

**Remarks:**

**⋄ Integers 1 , 2 , 3 , … are called positive integers or natural number and denoted by I ^{+} or N.**

**⋄ Integers …, −3, −2, −1 are called negative integers and denoted by I ^{–}**

**⋄ Integers 0 , 1 , 2 , 3 , … are called whole numbers or non-negative integers.**

**⋄ Integers …, −3, −2, −1, 0 are called non-positive integers**

**Rational Numbers:**

**Rational Numbers:**

**The numbers which can be expressed in the form p/q where p and q are integers, H.C.F. of p and q is 1 and q ≠ 0, are called rational numbers and their set is denoted by Q.**

**Thus Q = { p/q : p , q ∈I and q ≠0 and HCF of p , q is 1 }.**

**It may be noted that every integer is a rational number since it can be written as p/1.**

**It may also be noted that all recurring decimals are rational numbers. e.g., p = = 0.33333…**

**Then, 10p − p = 3**

**=> p = 1/3 , which is a rational number.**

**Irrational Numbers:**

**Irrational Numbers:**

**There are numbers, which can not be expressed in p/q form. These numbers are called irrational numbers and their set is denoted by Qc (i.e. complementary set of Q)**

**e.g. √2 , 1 +, √3 , π , e , √5 etc.**

**Irrational numbers can not be expressed as terminating decimals or recurring decimals.**

**Real Numbers:**

**Real Numbers:**

**The complete set of rational and irrational numbers is the set of real numbers and is denoted by R. Thus R = Q ∪ Q ^{C}**

**It may be noted that N⊂ I⊂ Q⊂ R. The real numbers can also be expressed in terms of the positions of a point on the straight line.**

**The straight line is defined as the real number line wherein the position of a point relative to the origin (i.e. 0) represents a unique real number and vice versa.**