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Set

A set is a collection or a group of well – defined objects or the items having some common characteristics.
Well defined means – Meaning should be clear whether a particular object belong to the group or not.
Eg: Well – defined examples:

  • The set of students of class VIII
  • The set of even numbers between 1 and 10.
  • The set of teachers of a school.

Not well – defined examples:

  • The set of tall boys of a school.
  • The set of good girls of a class.

Notation

  • Usually sets are denoted by capital letters and its elements or members are denoted by small letters or numbers or names.
  • The elements or members of sets are enclosed in the braces { }.
    Eg: A = {a, e, i, o, u},
    N = {1, 2, 3, 4, . . . }
    W = {0, 1, 2,  . . . . } etc.
  • The member ship of a set is denoted by the symbol '\(\in\)' and the non – membership is denoted by \(\notin\).
    Eg: If 'a' is an element of M, then a \(\in\) M and read as 'a' belong to M.

Specification of Set – Methods of describing sets
There are three methods of describing the sets.

  • Listing Method
    In this method the elements are listed within braces { }. The elements are separated with common. This method is also called as Tabulation or Roster method.
    Eg: W = {0, 1, 2, 3, 4, . . . . }
    A =  {a, e, i, o, u}
  • Description Method
    In this method some descriptive phrases are use enclosing within the braces { }.
    Eg: A = {prime number less than 10}
    B = {First five letters of English alphabet}
  • Set Builder Method
    In this method, all the elements of the sets are used by a variable stating the property or properties satisfied by the element of the set.
    Eg: W = { x : x is whole number less than 10]
    Which is read as W is set of all x such that x is a whole number less than 10.
    The symbol (:) or (/) after variable that denote member stand for such that.

Types of Sets

  1. Empty Set
    A set containing no elements is called an empty set or Null set or Void set. It is denoted by \(\phi\)(phi) or { }.
    Eg: The set of natural number between 9 and 10.
    The set of prime number 19 and 22.
  2. Singleton Set / Unit Set
    A set containing only one element is known as singleton set or Unit set.
    Eg: The set of present president of a country.
    A = {2}
    B = {Odd number between 6 and 8}
    C = {x : x is a prime number between 18 and 12}
  3. Finite Set
    A set containing finite (countable) number of elements is known as finite set.
    Eg: A = {2, 4, 6, 8, 10}
    B = {x : x is prime number less than 10}
  4. Infinite Set
    A set containing infinite (Uncountable) number of elements is known as infinite set.

Cardinal Number of Sets / Cardinality of Set
The number of elements or members present inside the set is known as cardinal number or the cardinality of the set. It is denote by n( ).
Eg: If A = {2, 4, 6, 8, 10}
n(A) = 5 i.e. cardinal number of A is 5.

Relation between Set

  1. Equal Sets
    Two sets are said to be equal sets if they contain exactly same elements.
    Eg: A = {1, 3, 5, 7, 9}, B = { 3, 7, 5, 9, 1} i.e. A = B
    A = {4, 6, 8, 10, 12}, B = {x : x = y + 1, y \(\in\) odd number 4 \(\leq\) y \(\leq\) 11
  2. Equivalent Sets:
    Two sets are said to be equivalent sets if they contain same number of members (elements) in the sets, i.e. the cardinal number are same.
    Eg: A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}
    A equivalent B because n(A) = 5, n(B) = 5 and written as A \(\sim\) B or A \(\leftrightarrow\), i.e. '\(\sim\)' or '\(\leftrightarrow\)' denotes equivalent sets.
  3. Overlapping Sets
    Two sets are said to be overlapping sets if they have at least one element in common.
    Eg: A = {a, b, c, d, e} B = {a, e, i, o, u}
    The common elements of A and B are a and e \(\Rightarrow\) A and B are overlapping.
  4. Disjoint Sets
    The sets are said to be disjoint if they do not have any common elements.
    Eg: A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}
    A and B are disjoint sets

Subsets
In two sets A and B, the set A is said to be sub set of B if every element of set A are the elements of set B.
Eg: If A = {factor of 6}
= {1, 2, 3, 6}
B = {factor of 12}
= {1, 2, 3, 4, 6, 12}
So, A is the subset of set B and it is written as A \(\subseteq\) B.
\(\therefore\) The symbol '\(\subseteq\)' denote subsets.

NOTE:  
Empty set is subset if every set \(\phi\) \(\subset\) A, \(\phi\) \(\subset\) B.
Every set is subset of itself i.e. A \(\subseteq\) A, B \(\subseteq\) B

Proper Subset
The subject A of set B is called proper subset if there number at least one element which contain in A.
Eg: B = {2, 4, 6, 8, 10}, A = {2, 4, 8, 10}
A \(\subset\) B (A is proper subset of B)

Improper Subsets
The subset A of the set B is said to be improper subset if every elements of B are also the elements of A.
Eg: A = {2, 4, 6, 8, 10} B = {2, 6, 4, 8, 10}
A \(\subseteq\) (A is improper subset of B)
Improper subset is also the equal set.

Superset
In two sets A and B if A is the subset of B then B is the super set of A. It is written as A \(\subseteq\) B (A is subset of B)
B \(\supseteq\) A (B is superset of A)

Universal Set
The set under consideration, which contains all the elements of other sets. It is denoted by \(\cup\).
If A = {2, 4, 6, 8}, B = {2, 3, 4, 5} and C = {2, 3, 5, 7}
U = {1, 2, 3,  . . . 10}

Number of Subsets
Number of subsets can be formed according to the number of elements present in the sets. By using simple formula:
i.e. number of subsets = 2n
where, n = number of elements in the sets
Eg: A = {1}
n(A) = 1
number of subsets = 21 = 2
i.e. { } {1}
B = {2, 4}
n(B) = 2
Number of subsets = 22 = 4
i.e. {2} {4} {2, 4} { }
\(\therefore\) Number of proper subsets = 2n – 1
where, n = number of elements in the sets

Venn diagram
The some geometrical figures like rectangle, circle or oval are used to represent the set or for the operation of set is known as venn diagram. Generally rectangle is use to denote Universal set. Circle and oval are used to represent other sets.

NOTE:
The concept was at first use by Swiss mathematician Euler but further it was denoted by British mathematician John Venn. So its name is Venn diagram.

Operation of Sets
Generally operation of sets includes:

  • Union of sets
  • Intersection of sets
  • Difference of sets
  • Complement of sets

Union of Sets B
If A and B are the two sets, the union of two sets A and B is the set which contains every element belonging either A or B or both A and B. Union is denoted by symbol 'cup'.
So, A union B \(\Rightarrow\) A \(\cup\) B = {x : x \(\in\) A or B}
(A \(\cup\) B) read as A union B or A cup B.
Eg: If A = {2, 4, 6, 8} and B = {4, 8, 12, 16}
A \(\cup\) B = {2, 4, 6, 8, 12, 16}

NOTE:
Common elements from both sets is written only once.

Intersection of Sets
If A and B are the two no empty sets, the intersection of the sets A and B is the set formed by taking only common elements of both sets. It is denoted by the symbol 'cap'.
The intersection of A and B means A \(\cap\) = {x : x \(\in\) A and x \(\in\) B}
(A \(\cap\) is read as A intersection B or A cap B.
Eg: If A = {2, 4, 6, 8} B = {4, 8, 12, 16}
A \(\cap\) B = {4, 8}

NOTE:
If there is no common between two sets then A \(\cap\) B = { } or \(\phi\)
Eg: If A = {2, 3, 5, 7}, B = {2, 4, 6, 8} and C = {1, 2, 3, 4, 5}
A \(\cap\) B \(\cap\) C = {2}

Difference of two Sets
The difference of two sets A and B is the set of all the elements present only in A but not in B. It is denoted by A – B.
A-B = {x : x \(\in\) A but x \(\notin\) B}
B-A = {x : x \(\in\) B but x \(\notin\) A}
If A = {2, 3, 5, 7} B = {2, 4, 6, 8}
A – B = {3, 5, 7}
Similarly B – A = {4, 6, 8}

Complement of Sets
The complement if any set is the sets of all elements in the Universal set except the elements of the given set.
The complement of A is denoted by \(\bar{A}\) or A' or Ac
Eg: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 4, 6, 8, 10}
\(\bar{A}\) = U – A = {1, 3, 5, 7, 9}
\(\bar{A}\) = {x : x \(\in\) U, but x \(\notin\) A}
Venn diagram of Different Sets

( FIG)

Some Examples:
If A = {1, 2, 3, 4, 5,}, B = {2, 3, 5, 7, 11} and C = {3, 5, 7, 9} find:

  1. A \(\cup\) B – C
  2. A \(\cup\) B \(\cup\) C
  3. A \(\cap\) B
  4. A \(\cap\) B \(\cap\) C
  5. A – B
  6. A – C
  7. (A \(\cup\) B) – C
  8. C – B illustrate them in Venn diagram

Solution:
Here,

  1. A = {1, 2, 3, 4, 5}   B = {2, 3, 5, 7, 11}   C = {3, 5, 7, 9}
    A \(\cup\) = {1, 2, 3, 4, 5} \(\cup\) {2, 3, 5, 7, 11}
    = {1, 2, 3, 4, 5, 7, 11}
  2. A \(\cup\) B \(\cup\) C = {1, 2, 3, 4, 5} \(\cup\) {2, 3, 5, 7, 11} \(\cup\) {3, 5, 7, 9}
    = {1, 2, 3, 4, 5, 7, 9, 11}
  3. A \(\cap\) B = {1, 2, 3, 4, 5} \(\cap\) {2, 3, 5, 7, 11}
    = {2, 3, 5}
  4. A \(\cap\) B \(\cap\) C = {1, 2, 3, 4, 5} \(\cap\) {2, 3, 5, 7, 11} \(\cap\) {3, 5, 7, 9}
    = {3, 5}
  5. A – B = {1, 2, 3, 4, 5} – {2, 3, 5, 7, 11}
    = {1, 4}
  6. A – C = {1, 2, 3, 4, 5} – {3, 5, 7, 9}
    = {1, 2, 4}
  7. (A \(\cup\) B) – C = {1, 2, 3, 4, 5} \(\cup\) {2, 3, 5, 7, 11} \(\cup\) – {3, 5, 7, 9}
    = {1, 2, 3, 4, 5, 7, 11} – {3, 5, 7, 9}
    = {1, 2, 4, 11}
  8. C –  B = {3, 5, 7, 9} – {2, 3, 5, 7, 11}
    = {9}

Cardinality Relation of Sets

1) When the sets are disjoint:

  1. n(A \(\cup\) = n(A) + n(B)
    n(\(\overline{A \cup B}\)) = n(U) – n(A \(\cup\) B)
  2. n(A \(\cup\) B \(\cup\) C) = n(A) + n(B) + n(C)

2) When sets are Overlapping

  1. n(A \(\cup\) B) = n(A) + n(B) – n(A \(\cap\) B) or n(A) + no(A) + no(B) + n(A \(\cap\) B)
    n(overline{A \cup B}\) = n(U) – n(A \(\cup\) B)
  2. n(A \(\cup\) B \(\cup\) C) = n(A) + n(B) + n(C)n- n(A \(\cap\) B) – n(B \(\cap\) C) – n(A \(\cap\) C) + n(A \(\cap\) B \(\cap\) C) or n(A \(\cup\) B \(\cup\) C) = no(A) + no(B) + no(C) + n­o(A \(\cap\) B) + no(B \(\cap\) C) + no(A\(\cap\) C) + n(A \(\cap\) B \(\cap\) c)
    n(\(\overline{A \cup B \cup C}\)) = n(U) – n(A \(\cup\) B \(\cup\) C)
  3. NOTE:
    no(A) = only A
    no(A \(\cap\) C) = only A \(\cap\) C

Examples:

  1. A and B are the subsets of a universal set U, where n(U) = 75, n(A) = 38, n(B) = 40 and n(A \(\cap\) B) = 8
    (i) Represent the information in Venn – diagram
    (ii) Find n(\(\overline{A \cup B}\))
    Solution:
    Here,
    n(U) = 75
    n(A) = 38
    n(B) = 40
    n(A \(\cap\) B) = 8
    (i) Representation of data in Venn diagram
    FIG
    (ii) From Venn diagram , we have
    or, 30 + 8 + 32 + x = n(U)
    or, 70 + x = 75
    or, x = 5
    \(\therefore\) n(\(\overline{A \cup B}\)) = 5
    Alternative Process (By using formula)
    n(\(\overline{A \cup B}\)) = n(U) – n(A \cup B)
    = 75 – [n(A) + n(B) – n(A \(\cap\) B)]
    = 75 – {38 + 40 – 8}
    = 75 – 70
    = 5 
    NOTE: Any method can be followed

Exercise

  1. If A = {a, b, c, d, e, f}, B = {b, c, d, f, g} and C = {a, e, i, o, u}. Find
    (i) A \(\cap\) B
    (ii) B \(\cap\) C
    (iii) A \(\cup\) B
    (iv) (A \(\cup\) C) \(\cap\) B
  2. If the universal set U = {1, 2, 3 . . . . 10} and A = {2, 4, 6, 8}, B = {1, 3, 5, 7, 9} and C = {2, 3, 5, 7} are the set of U. Find:
    (i) \(\bar{A}\)
    (ii) (\(\overline{A \cup B}\))
    (iii) (\(\overline{A \cap C}\))
    (iv) (\(\overline{A \cup B}\)) \(\cup\) C
    (v) (A – B) \(\cap\) C
  3. If P and Q are the subsets of a Universal set U. If n(U) = 80, n(P) = 36, n(Q) = 45 and n(\(\overline{P \cup Q}\) = 58. Find:
    (i) n(P \(\cap\) Q)
    (ii) n(\(\overline{P \cup Q}\))
    (iii) no(P)
    (iv) no(Q)
  4. In an examination, 40 students passed in English, 55 passed in Nepali and 5 students failed in both the subjects. If 80 students appeared in the examination, how many of them passed in both the subjects.
  5. In a survey of 95 students, 40 like basketballs, 55 like football and 10 students like neither football nor basketball. Find:
    (i) How many students like both the games?
    (ii) How many students like only basketball?
    (iii) How many students like only football?
  6. Write the set notation for the shaded part in the following Venn diagram. 

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