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**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:

**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.**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.**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:

**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.**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}**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}**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**

**Equal Set**Two sets are said to be equal if elements on them are exactly the same.

E.g.**Equivalent Sets**

Two sets are said to equivalent if the number of members of set are equal.

E.g.

i.e. n(A) = n(B) or, A \(\sim\) B**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}**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}

**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**

Possible subsets are: {a} {b} {c} {a, b} {b, c} {a, c} {a, b, c} { }

**Exercise **

- 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} - Write all the possible subsets of:

a) {1, 2, 3}

b) {5, 0}

c) {i} - Form the universal set for:

a) {students of class six}

b) {boys of class 6}

c) {cabbage, cauliflower, potato} - 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} - Write the overlapping elements separately from the following:

**Set Operations**

There are four fundamental set operations. They are:

- Union of Sets
- Intersection of Sets
- Difference of Sets
- 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.]

**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}

**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}

- A \(\cap\) = {1, 2, 3, 4, 6, 12} \(\cap\) {1, 2, 4, 5, 10, 20}

\(\therefore\) A \(\cap\) B = [1, 2, 4} - 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}

**Exercise**

- From the given diagram find

b) A \(\cap\) B - If A = {factors of 20} and B = {factors of 24}. Find:

i) A \(\cup\) B

ii) A \(\cap\) B

Also, illustrate in diagram. - 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. - If A = {a, b, c, d, e} and B = {a, e, i, o, u}, find A \(\cup\) B.
- If A = {a, e, i, o, u} and B = {a, b, c, d, e}, find A – B and B – A.
- If A = {a, b, c, d, e} and B = {a, e, i, o, u}, find A \(\cap\) B and show in venn diagram.
- If A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 4, 6, 8}, find A \(\cap\) B.
- If A = {whole number less than 10} and B = {add number less than 9}, find A \(\cap\) B and A \(\cup\) B.
- 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.
- From the following figure find A \(\cup\) B, B \(\cap\) A, B – A and A – B.

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