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SETS THEORY

George Cantor the German mathematician developed the theory of Set during 1874 to 1884. Nowadays the set theory is used in almost every branches of mathematics. Set theory is one of the branches of modern algebra. A set can be defined as a collection of well-defined and well distinguished objects.By “ well-defined ” we mean that it must be possible to tell about objects without any doubt. For example the collection of students is a well-defined collection while the collection of good student is not a well-defined collection because good students are vague (not clear). All sets are denoted by capital English alphabets e.g. A, B, C, X, Y, Z etc. The following are some common examples of set.

  1. The collection of capital English alphabets is a set containing the elements from A to Z.
  2. The collection of prime numbers less than 30.
  3. The collection of vowels of English alphabet.
  4. The collection of natural numbers less than 20 which are divisible by 3.
  5. The collection of all states of India.
  6. The collection of rivers of India.
  7. The collection of IPL cricket teams.

Examples of Sets

Examples of some standard sets

  1. N denotes set of all natural numbers.
  2. W denotes set of all whole numbers.
  3. Z denotes set of all integers.
  4. Q denotes set of all rational numbers.
  5. R denotes set of all real numbers.
  6. C denotes set of all complex numbers.
  7. Z+, Q+ and R+ denotes set of all positive Integers, Rational numbers and Real numbers respectively.
  8. Z, Q and R denotes set of all negative Integers, Rational numbers and Real numbers respectively.

Element

In the definition of set the word object which constitute the set are called elements are members of the set.Elements of a set are denoted by small letters for example a, b, c, x, y, z.Suppose p is an element of a set A. Then we write p∈A and read it as p belongs to A. If  p is not an element of set A then we write p∉ A read it as p does not belong to A.

Representation of a set

Any set can be represented in the following two ways. We can use any one of these two methods according to our convenience.

  1. Roster method
  2. Set builder method

Roster method

In this method, the elements of a set are written within curly braces. And elements are separated by commas. Example a set of natural numbers less than 10 will be described as {1, 2, 3, 4, 5, 6, 7, 8, 9}. Similarly asset of English vowels can be described as {a, e, i, o, u}.

Note: we can write the elements of a set in any order. Thus  { 1, 2, 3, 4, 5}, { 3, 1, 5, 4, 2} and {5, 4, 3, 2, 1} denote the same set.

Set Builder method

In this method set is described by specifying the common properties of all elements. Thus standard format to write a set in this method is A = { x : P(x) is true } or A = { x | P(x) is true }. Where x is an arbitrary element and P(x) is the common property of elements. The symbols ‘ | ‘ and ‘ : ‘ are read as such that. For example a set of integers greater than 5 and less than equal to 15 can be written as A = { x : x is an Integer and x < 5 ≤ 15 }.

Types Of Sets

Depending upon the number of elements present in a set. Sets can be classified as finite set, infinite set, Empty set and Singleton set.

Finite Set

If a set has a finite number of elements then the set is said to be a finite set. The number of elements present in a finite set is called the cardinal or order of the finite set. Example of a finite set is A = { 1, 2, 3, 5, 9, 7, 15, 19 }. The cardinal or order of set A is 8 because it contains only 8 elements.

Examples of Finite Sets

  1. If A = {x : x ∈ N and x < 30} will contain natural numbers from 1 to 29. Since there are 29 elements in A, thus it a finite set.
  2. Set of letters of English Alphabets is a finite set. And it will contain 26 letters from a to z.
  3. A = {x : x ∈ Z and 2 ≤ x ≤ 10} contains 9 Integers from 2 to 10. Thus it a finite set.

Infinite Set

If a set has infinite number of elements then the set is called an Infinite set. For example a set of natural number is an Infinite set because there is no end of natural numbers. A set of all points or lines on a plane is also an example of Infinite set.

Examples of Infinite Sets

  1. The Set A = {x : x ∈ N and x > 100} is an Infinite set because there are infinite natural numbers greater than 100. Thus it is an Infinite set.
  2. B = {x : x ∈ Z and x < 5} contains infinite Integer elements from 5 to negative infinite Integer.
  3. Set of concentric circles in a plane is set an Infinite set.
  4. C = {x : x ∈ R and 0 < x < 1} is also an infinite set. Because set C contains infinite real numbers between 0 and 1.

Empty Set

If a set has no element then the set is called empty or null or void set. An empty set is denoted by Φ.

Examples of Empty Sets

  1. A = {x : x is a real number and x2 + 1 = 0} is an empty set because the values of x, after solving x2 + 1 = 0, are -√-1 and +√-1 which are not real number.
  2. A = {x : x2 = 5, x ∈ N} is an empty set because values of x will be √5 and -√5 which are not natural numbers.
  3. A = {x : x2 + 3x + 2 = 0, x ∈ N} is an empty set because after solving the quadratic equation, the roots of equations are -1 and -2. Since -1 and -2 are not natural numbers therefore it is an empty set.
  4. A = {x : x is a point common to any two parallel lines} is an empty set because two parallel lines can not have a common point.

Singleton Set

If a set has only one element then the set is called singleton set. Example of singleton set are A = { 2 }, B = { d } and C = { Φ }.

Except these sets there are some other sets also. These sets are Subset, Power set, Universal set, Equivalent set, Equal sets and Comparable sets.

Examples of Singleton Sets

  1. A = {x : x ∈ N, |x| = 6} is a singleton set because x belongs to natural number. And |x| = 6 has two values 6, -6. But -6 is not a natural number thus the solution set will have only 6.
  2. B = {x : x is an Even Prime number} is a singleton set. Because 2 is the only Even Prime number among Prime numbers.
  3. C = {x : x ∈ N and x2 – 1 = 0} is singleton set. Because x2 – 1 = 0 has two solution 1 and -1. Since -1 is not natural number thus the solution set will contain only 1. Hence set C is a singleton set.

Subset

If A and B are two sets and each element of set A is also an element of Set B then A is called subset of B. If A is subset of B then we write A ⊆ B and it is read as A is the subset of B. mathematically A ⊆ B can be expressed as. if a ∈ A then a ∈ B. if A is subset of B then B is called the super set of A and it is denoted as B ⊇ A. A subset can be anyone of two types and these are as:

Improper Subset

Every set is a subset of itself and the Empty set is subset of every set. These two subsets are called Improper subsets. For example if A ⊆ A then A and Φ are Improper subsets.

Proper Subset

If and B are two sets and if A ≠ B and A ⊆ B then A is the Proper subset of B and it is written as A ⊂ B.

Theorems on Subset

  1. Every set is a subset of itself. For example if A is a set then each element of A is clearly in A itself, thus A ⊆ A.
  2. Empty Set (Φ) is the subset of every set. if A is a set and Φ be the empty set then Φ ⊆ A.
  3. If A be a subset of B and B be a subset of C then A is a subset of C also. Mathematically if A ⊆ B and B ⊆ C then A ⊆ C.
  4. The total number of subset of a finite set containing n elements is 2n. For example if a set has 3 elements then number of subset is 8.
  5. The number of Proper subset of a finite set containing n element is 2n – 2.
  6. N ⊂ Z ⊂ Q ⊂ R ⊂ C, where N denotes natural numbers, Z denotes Integers, Q denotes rational numbers, R denotes Real numbers and C denotes complex numbers.

Power Set

The set of all possible subsets of a set is called power set. if A be a set then power set of A is denoted by P(A). Mathematically it is denoted as P(A) = { S : S ⊂ A }, where S denotes a subset of A.
If A = {1, 2, 3} then
P(A) = {Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}, {1, 2, 3}}
If a finite set has n elements then its power set will have 2subsets.

Universal Set

The containing all possible values in a given context is called universal set. A universal set is denoted by U. Examples of Universal set are as:

  1. A set of Real numbers is a Universal set.
  2. A set of all points in a plane is an example of Universal set.

Equivalent Set

Two or more sets are said to be equivalent sets if each set has same number of elements or same order or cardinal number. if A = {a, b, c} and B = {1, 2, 3} then A and B are equivalent sets because each of them has 3 elements . Mathematically it can be denoted as n(A) = n(B), where n(A) is order of A and n(B) is order of B.

Equal Set

If A and B are said to be equal set if each element of A is also an element of B and Each element of B is also an element of A. Mathematically if A ⊂ B and B ⊂ A then A = B. For example if A = {1, 2, 3, 4, 5} and B = {5, 4, 3, 2, 1} then A and B are equal sets.

Comparable Sets

Two sets are said to be comparable if either A is a proper subset of B or B is proper subset of A. Mathematically it can be written as A ⊂ B or B ⊂ A. For example if A = {1, 2, 3} and B = {1, 2, 3, 4, 5} then A ⊂ B. Hence A and B are comparable sets.

Disjoint Sets

If two sets A and B have no common element then A and B are called Disjoint Sets. For example A = {a, b, c} and B = {1, 2, 3} are disjoint set because they have no any common element. Mathematically, let A and B are two sets then A ∩ B = Φ, where Φ denotes empty set.

Operations on Sets

Union of Sets

If A and B are two sets then the union set of A and B will contain the all elements of both sets A and B. The union of two sets A and B is denoted by A ∪ B. Mathematically union of two sets can be written as if a ∈ (A ∪ B) then a ∈ A or a ∈ B. Or it can be written as A ∪ B = {a : a ∈ A or a ∈ B}. if a is not an element of A ∪ B then we write is as if a ∉ (A ∪ B) then a ∉ A and a ∉ B.

Example 1: If A = {7, 8, 9, 10} and B = {1, 2,3, 4, 5, 6} then A ∪ B = {1, 2, 3, 4, 5, 6, 7, , 9, 10}.

Example 2: if A = {a : a = 2n + 1, n ∈ Z} and B = {a : a = 2n, n ∈ Z} then find A ∪ B.
Since A represents a set of odd Integers and B represents a set of even Integers. Thus the union of these two sets will represent a set of Integers. Hence A ∪ B = {a : a is an Integer} or A ∪ B = Z, where Z denotes set of Integers.

Theorems on Union

  1. If A and B are two sets and A ⊂ B then A ∪ B = B and if B ⊂ A then A ∪ B = A.
  2. If A be a set and Φ be the empty set then A ∪ Φ = A or Φ ∪ A = A. This property is called Identity Law of Union.
  3. If A is a set then A ∪ A = A and A ∪ U = U. This property is called Idempotent Law of Union.
  4. If A and B are two sets then A ∪ B = B ∪ A, so Union is commutative.
  5. If A, B and C are three sets then A ∪ (B ∪ C) = (A ∪ B) ∪ C. This is the associative law of union.
  6. If A, B and C are three sets then A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). This is the distributive property of union.
  7. If A be a set then A ∪ A’ = U, where A’ denotes complement of A and U is universal set.
  8. If A and B are two sets. If A has m elements and B has n elements then the minimum number of elements in A ∪ B will be as: when n > m then n elements, When n < m then m elements, and if m = n then m or n elements.
  9. If A and B are two non-disjoint and non-void sets then n(A ∪ B) = n(A) + n(B) – n(A ∩ B). If They are disjoint then n(A ∪ B) = n(A) + n(B).
  10. If A, B and C are three 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).
  11. n(A’ ∪ B’) = n(U) – n(A ∩ B), Where U is the universal set.

Intersection of Sets

The Intersection of two sets is the set of common elements of both sets. If A and B are two sets then Intersection of A and B is denoted by A ∩ B. Mathematically if a ∈ (A ∩ B) then a ∈ A and a ∈ B. In set Builder method it is expressed as A ∩ B = {a : a ∈ A and a ∈ B}. If a ∉ (A ∩ B) then a ∉ A or a ∉ B.

Example: if A = {a, b., c, d, e} and B = {a, e, i, o, u} then A ∩ B = {a, e}.

Theorems on Intersection

  1. If A is the proper subset of B (A ⊂ B) then A ∩ B = A and if B is proper subset of A (B ⊂ A) then A ∩ B = B.
  2. If Φ be the empty set then A ∩ Φ = Φ or Φ ∩ A = Φ. This is the Identity law of Intersection.
  3. If A be any set then A ∩ A = A. And A ∩ U = A where U is the Universal set. This is the Idempotent law of Intersection.
  4. Intersection of two sets are commutative. If A and B are two sets then A ∩ B = B ∩ A.
  5. If A, B and C are three sets the (A ∩ B) ∩ B = A ∩ (B ∩ C). This is the associative law of intersection.
  6. If A, B and C are three sets then A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). This is the distributive property of intersection.
  7. If A be any set then A ∩ A’ = Φ, where Φ is empty set and A’ denotes complement set of A.
  8. If sets A and B have m and n elements respectively then maximum number of elements in A ∩ B will be as: when n > m then m elements, When n < m then n elements, and if m = n then m or n elements.
  9. n(A’ ∩ B’) = n(U) – n(A ∪ B), Where U is the universal set.

Complement of Set

If U be the universal set and A be any other set then the complement of set A denoted by A’ or Ac is the set of all those elements which are not present in A. Thus it can be written as A’ = {x : x ∈ U and x ∉ A}. If a is an element then and a ∈ A’ then a ∈ U but a ∉ A.

Example: If U = {x : x is an Integer} and A = {x : x is negative Integers} then A’ = {x : x is positive Integers}.

Example: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9} then A’ = {2, 4, 6, 8, 10}.

Theorems on Complement Operation

  1. If U be the universal set then complement of U will be the Empty set. Mathematically, U’ = {x : x ∈ U and x ∉ U} = Φ.
  2. If Φ be the Empty set then complement of Φ is the universal set. Mathematically, Φ’ = {x : x ∈ U and x ∉ Φ} = U.
  3. The complement of complement of a set is the set itself. Mathematically if A be any set then (A’)’ = A.
  4. If A and B are two sets then (A ∪ B)’ = A’ ∩ B’ and (A ∩ B)’ = A’ ∪ B’. These two theorems are known as De Morgans Theorem.
  5. If A and B are two set and A is the subset of B (A ⊆ B) then complement of B is the subset of complement of A (B’ ⊆ A’).

Difference of set

Let A and B are two sets then the difference of A and B, denoted by A – B, is the set of elements of set A which are not present in B. Mathematically A – B is denoted as: A – B = { x : x ∈ A and x ∉ B}.

Similarly, B – A is the set of elements of set B which are not present in A. Mathematically, B – A = {x : x ∈ B and x ∉ A}.

Example: If A = {3, 5, 7, 2, 9} and B = {1, 2, 3, 4, 5}  then A – B = {7, 9} and B – A = {1, 4}.

Theorems on Difference Operation

If A, B and C are three sets then

  1. A – B = A ∩ B’
  2. B – A = B ∩ A’
  3. A – B = A and B – A = B if A ∩ B = Φ
  4. (A – B) ∪ B = A ∪ B
  5. (A – B) ∩ B = Φ
  6. (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)
  7. A – (B ∪ C) = (A – B) ∩ (A – C)
  8. A – (A ∩ C) = (A – B) ∪ (A – C)
  9. A ∩ (B – C) = (A ∩ B) – (A ∩ C)

Symmetric Difference of Sets

Let A and B are two sets then their symmetric difference is the union of the sets (A – B) and (B – A). Or Symmetric difference of two sets is a set of elements which are not common to both the sets. Symmetric difference of A and B is denoted by A Δ B.Thus Mathematically we write A Δ B = {x : x ∈ (A – B) or x ∈ (B – A)}.

Example: If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A Δ B = {1, 3, 5, 8}.

Theorems on Symmetric Difference

If A, B and C are three sets then

  1. A Δ B = (A ∪ B) – (A ∩)
  2. A Δ Φ = A, where Φ is the empty set.
  3. A Δ A = Φ
  4. A Δ B = Φ if A = B
  5. A ∩ (B Δ C) = (A ∩ B) Δ (A ∩ C)

Cartesian Products of Sets

If A and B are two non-empty sets, then set of all ordered pair (a, b) such that a ∈ A and b ∈ B is called the cartesian product of A and B. The cartesian product A and B is denoted by A x B. Mathematically we write A x B = {(a, b) : a ∈ A and b ∈ B}.

Ordered Pair

An ordered pair consists of two elements in a given fixed order. Example of an order pair is (2, 5) where 2 belongs to a set A and 5 belongs to a set B.

Equality of Ordered Pairs

if (a1, b1) and (a2, b2) are two ordered pair then they will be equal if a1 = a2 and b1 = b2. Thus we can write (a1, b1) = (a2, b2) ⇔ a1 = a2 and b1 = b2.

Example: If A  = {1, 2, 3} and B = {4, 5} then A x B = {(1, 4), (1, 4), (2, 4), (2, 5), (3, 4), (3, 5)}. Similarly B x A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)}.

Properties of Cartesian Product

If A and B are two sets then A x B is not equal to B x A. Thus A x B ≠ B x A. A x B and B x A will be equal if and only if A  = B.

For any two sets A and B (A x B) ∩ (B x A) is an Empty set. Thus (A x B) ∩ (B x A) = Φ.

If A, B and C are three sets then

  1. A x (B ∪ C) = (A x B) ∪ (A x C)
  2. A x (B ∩ C) = (A x B) ∩ (A x C)
  3. A x (B – C) = (A x B) – (A x C).
  4. A x (B’ ∪ C’)’ = (A x B) ∩ (A x C)
  5. A x (B’ ∩ C’)’ = (A x B) ∪ (A x C)

If A, B ad C are three sets and A ⊆ B then A x C ⊆ B X C.

If A, B, C and D are four sets and if A ⊆ B and C ⊆ D then A x C ⊆ B x D.

If A and B are two sets having m and n elements respectively then the number of ordered pair in A x B will be m x n. And The number of subsets of A x B will be 2mn.

If A, B and C are three sets and A x B = A x C then B = C.

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