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Set

Set
A collection of well-defined objects is called set.
E.g.: \(\left.\begin{array}{l}\text{Group of domestic animals}\\ \text{Group of colours}\\ \text{Collection of two wheelers} \end{array} \right\}\) well defined

\(\left.\begin{array}{l}\text{Collection of beautiful girls}\\  \text{Group of intelligent boys} \end{array} \right\}\) not, defined, so it is not set

Set is inclosed inside the curley brackets. It's members are denoted by small letters and set is denoted by capital letters.
E.g.:
A = {a, b, c, d, e}
B = {a, e, I, o, u}
A \(\Rightarrow\) set, {} – curly brackets
a, e, i, o, u \(\Rightarrow\) members of set

Note:
\(\in\) \(\Rightarrow\) belongs to
\(\notin\) \(\Rightarrow\) does not belong to
E.g. a \(\in\) A, I \(\in\) B and u \(\notin\) A
i.e. 'a' belongs to set A and 'u' does not belong to set A.

Exercise

  • Which of the following represents a set?
    • collection of colours of rainbow
    • collection of cooking numbers between 1 to 10
    • collection of high mountains of Nepal
    • collection of handsome boys
    • collection of fat girls
  • Let F = {Factors of 8}, M = {Multiple of 2 less than 9} then insert \(\in\) or \(\notin\) in the blanks.
    • 3 ______ F 
    • 5 ______ M
    • 4 ______ F
    • 6 ______ M

Methods of Representing Sets
Sets are represented by different methods. They are:

  1. Listing Method
    The elements of a set are listed or tabulated.
    E.g. A = {a, e, i, o, u} which is described by listing all vowels of English Alphabets.
  2. Description Method
    The elements of sets are indicated by its description.
    E.g. A = {days of a week} or A is the set of days of a week
    X = {vowel of English alphabets} or X is the set of vowels of English alphabets.
  3. Set Builder Method
    The elements of sets are described by property of sets in form of set.
    E.g. A = {x:x is a vowel of English alphabets}
    A is a set of all x such that x is a vowel of English alphabets.
    B = {a: a is a counting number less than 10}
    B is a set of all a such that a is a counting number less than 10.

Some examples
Write as indicated in the bracket.

  • The set of counting numbers between 3 and 15. [in listing method and set builder method]
    \(\Rightarrow\) A = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} – Listing method
    \(\Rightarrow\) A = {x:x is counting number between 3 and 15} – Set builder method
  • {x:x is an odd number between 2 and 8} [listing and description method]
    \(\Rightarrow\) A = {3, 5, 7) – Listing method
    \(\Rightarrow\) A = {collection of odd numbers between 2 and 8} – Set builder method

Exercise

  • Write down the following sets in set builder form.
    • A = {4, 8, 12, 16}
    • B = {1, 2, 3, 4, 5}
    • A = The set of girls students of your class
    • B = The set of prime numbers less than 25 and more than 10.
    • A = The set of months of English calendar.

Types of Sets
There are different types of set:

  1. Empty/ Null/ Void Set
    A set having no elements is known as empty/ null/ void set. It is denoted by \(\phi\) or { }.
    E.g. The set of present king of Nepal.
    The prime number between 7 and 10.
  2. Singleton/ Unit Set
    A set containing only one member or element is known as singleton set.
    E.g. The set of present prime minister of Nepal.
    A = {e}
  3. Finite Set
    A set containing finite (countable) number of elements is known as finite set.
    E.g. A = {collection of even numbers less than 10}
    B = {vowels of English alphabet}
  4. Infinite Set
    A set containing infinite (uncountable) number of elements is known as infinite set.
    E.g. A = {collection of Natural Numbers}
    B = {4, 8, 12, 16, ……}

Cardinal Number of Set
The number of elements present in the set is known as cardinal number. It is denoted by n( ).
E.g. A = {2, 4, 6, 8, 10}
or, n(A) = 5
i.e. The cardinal number of set A is 5.

Set Relation

  1. Equal Set
    Two sets are said to be equal if elements on them are exactly the same.
    E.g.
    Set - 001
    A and B are equal sets.
  2. Equivalent Sets
    Two sets are said to equivalent if the number of members of set are equal.
    E.g.
    Set - 002
    Set A is equivalent to set B or A and B are equivalent.
    i.e. n(A) = n(B) or, A \(\sim\) B
  3. Overlapping Sets
    Two sets are said to be overlapping if there is at least one element common in them.
    E.g. A = {a, e, i, o, u}
    B = {a, b, c, d, e}
    set - 003
    A and B are overlapping sets.
  4. Disjoint Sets
    Two sets are said to be disjoint if there is no common element (members) in between them.
    E.g. A = {1, 3, 5, 7, 9}
    B = {2, 4, 6, 8, 10}
    Set - 004
    A and B are disjoint.

Subsets
A set is subset of another set if its every elements lies on the another set.
E.g. if A = {2, 4, 6, 8, 10}
B = {2, 6, 8}
B \(\subset\) A

Universal Sets
Let U = {1, 2, 3, ………, 10} \(\rightarrow\) Universal Set
A = {1, 3, 5, 7, 9}
B = {2, 4, 6, 8, 10}
C = {2, 3, 5, 7}
A \(\subset\) U, B \(\subset\) U and C \(\subset\) U
The set under consideration which contains all the elements of other sets known as universal set.
So, all the other sets are subsets of universal set.

Examples
Write the possible subsets of A = {a, b, c}
Solution:
Possible subsets are: {a} {b} {c} {a, b} {b, c} {a, c} {a, b, c} { }

Exercise

  1. State whether the following are equivalent or equal sets.
    a) A = {natural number less than 6} and B = {first five counting numbers}
    b) A = {first five multiples of 3} and B = {first five multiples of 5}
  2. Write all the possible subsets of:
    a) {1, 2, 3}
    b) {5, 0}
    c) {i}
  3. Form the universal set for:
    a) {students of class six}
    b) {boys of class 6}
    c) {cabbage, cauliflower, potato}
  4. List the elements of given sets and state whether they are overlapping or disjoint.
    a) A = {factors of 5} and B = {factors of 3}
    b) A = {Multiples of 6 less than 20} and B = {Multiples of 4 less than 20}
  5. Write the overlapping elements separately from the following:
    Set - 01
    Set - 02

Set Operations
There are four fundamental set operations. They are:

  1. Union of Sets
  2. Intersection of Sets
  3. Difference of Sets
  4. Complement of Sets

But in this level, we will just read the union of two sets and intersection of two sets.

Union of two sets
The union of two sets A and B is the set of all elements which are present either in set A or set B or both the sets A and B. it is denoted by (A \(\cup\) B)
E.g. A = {1, 2, 3, 4, 5}
B = {2, 4, 6, 8}
A \(\cup\) B = {1, 2, 3, 4, 5, 6, 8}
[Note: Repeated letter is written only once.]

Union of sets

Intersection of two sets
The intersection of two sets A and B is the set of elements which are present both A and B. it is denoted by (A \(\cap\) B).
A = {2, 4, 6, 8, 10}
B = {2, 3, 5, 7}
A \(\cap\) B = {2}

Intersection

Example

If A = {factors of 12} and B = {factors of 20}, find:
(i) A \(\cap\) B
(ii) A \(\cup\) B

Solution:
A = {factors of 12} = {1, 2, 3, 4, 6, 12}
B = {factors of 20} = {1, 2, 4, 5, 10, 20}

  1. A \(\cap\) = {1, 2, 3, 4, 6, 12} \(\cap\) {1, 2, 4, 5, 10, 20}
    \(\therefore\) A \(\cap\) B = [1, 2, 4}
    Example - 01
  2. A \(\cup\) B = {1, 2, 3, 4, 6, 12} \(\cup\) {1, 2, 4, 5, 10, 20}
    \(\therefore\) A \(\cup\) B = [1, 2, 3, 4, 5, 6, 10, 12, 20}
    Example - 02

Exercise

  1. From the given diagram find
    Exercise - 01
    (I)
    Exercise - 02
    (II)
    Exercise - 03
    (III)
    a) A \(\cup\) B
    b) A \(\cap\) B
  2. If A = {factors of 20} and B = {factors of 24}. Find:
    i) A \(\cup\) B
    ii) A \(\cap\) B
    Also, illustrate in diagram.
  3. If A = {4, 8, 12, 16, 20, 24} and B = {6, 12, 18, 24}. Find:
    i) A \(\cup\) B
    ii) A \(\cap\) B
    Also, illustrate in diagram.
  4. If A = {a, b, c, d, e} and B = {a, e, i, o, u}, find A \(\cup\) B.
  5. If A = {a, e, i, o, u} and B = {a, b, c, d, e}, find A – B and B – A.
  6. If A = {a, b, c, d, e} and B = {a, e, i, o, u}, find A \(\cap\) B and show in venn diagram.
  7. If A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 4, 6, 8}, find A \(\cap\) B.
  8. If A = {whole number less than 10} and B = {add number less than 9}, find A \(\cap\) B and A \(\cup\) B.
  9. If A = {natural number less than 9} and B = {prime number upto 17}, find A \(\cap\) B, A – B, B – A and show the information in venn diagram.
  10. From the following figure find A \(\cup\) B, B \(\cap\) A, B – A and A – B.
    Set - 01
    Fig (a)
    Set - 02
    Fig (b)

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