Welcome to Edukum.com

A collection of well – defined objects is known as set. Well defined means whether particular object belong or does not belong to the set.

Eg:

- Collection of students of class seven \(\rightarrow\) Well defined – Set
- Collection of good students \(\rightarrow\) Not Well defined – Not Set
- Collection of intelligent students \(\rightarrow\) Not Well defined – Not Set
- Collection of capitals of SAARC countries \(\rightarrow\) Well defined – Set

**Notation **Generally set is denoted by capital letters. Its members or elements are denoted by small letters. The elements or elements are enclosed in the brackets { }.

Eg:

A = {a, e, i, o, u}, N = {1, 2, 3, 4, 5 . . . . . }

Eg:

A = {a, b, c, d, e} and B = {0, 1, 2, 3, 4, 5} and C = {2, 3, 5, 7, 11}

- a \(\in\) \(\rightarrow\) Read as a belongs to A
- g \(\notin\) A \(\rightarrow\) Read as g does not belongs to A.
- 3 \(\in\) \(\Rightarrow\) Read as 3 belongs to B

There are three ways of describing sets

- Listing method (roster method) (Tabulation method)
- Description method
- Rule or set builder method

Eg:

Listing or tabulation method:

N = {1, 2, 3, 4, 5} V = {a, e, i, o, u}

- Description method:

N = {Natural number less than 6}

V = {Vowels of English Alphabets} - Rule or Set builder method:

N = {x : x is natural number, less than 6}

V = {x : x is a vowel of English alphabet}

Read as: V is set of all x such that x is vowel of English alphabets.

Symbol (:) is such that or (/) \(\Rightarrow\) Such that

**Types of Sets **

**Empty/Null/Void Set**A set containing no elements is called an empty set. It is denoted by { } or \(\phi\) (phi)

Eg: Set of present king of Nepal

Set of male students of Padma Kanya school**Singleton Set**A set containing only one element or single elements is known as singleton set or unit set/

Eg: Set of present president of Nepal

Set of prime numbers between 2 and 4**Finite Set**A set containing limited or countable number of elements in the set is known as finite set.

Eg: A = {2, 4, 6, 8 . . . .}

B = {set of natural numbers}

**Cardinality of Sets (Cardinal Number)**The number of elements or the members present in the set is known as cardinal number sets.

Eg:

- V = {q, e, i, o, u}

n(V) = 5 - N = {1, 2, 3, 4, 5, 6, 7}

n(N) = 7

**Relation of Sets **Generally there are four types of set relations. They are:

**Equal Sets**Two sets A and B are said to be equal sets if they contain exactly the same elements in the set.

Eg: A = {2, 4, 6, 8, 10}, B = {6, 8, 2, 4, 10}

A and B are equal sets**Equivalent Sets**Two sets A and B are said to be equivalent sets if the number of member or the elements are equal in the given sets.

Eg: A = {a, e, I, o, u}, B = {2, 4, 6, 8, 10}

A \(\sim\) B read as A equivalent to B as the number of elements are same i.e. n(A) = n(B) \(\Rightarrow\) Cardinal number of set are same.**Overlapping Set**Two sets A and B are said to be overlapping if they have at least one element common in them.

Eg: A = {2, 4, 6, 8, 10}, B = {a, b, c, d, e}**Disjoint Sets**Two sets A and B are said to be disjoint, if they have no elements common in them.

E.g. A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}

**Subsets **The set which is formed by taking some elements from the given sets is known as subsets.

**NOTE:**Every set is subset of itself

Null set is subset of every set.

Eg: A \(\Rightarrow\) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B = {1, 3, 5, 7, 9}

C = {2, 4, 6, 8, 10}

C = {2, 3, 5, 7}

i.e.

B \(\subset\) A \(\Rightarrow\) is subset of A

A \(\supset\) B \(\Rightarrow\) A is super set of B

C \(\subset\) A \(\Rightarrow\) C is subset of A, i.e. all the elements are present in the set.

**There are two types of subsets:**

**Proper Subset**Eg: A = {1, 2, 3, 4, 5}, B = {2, 4, 5}

B \(\subseteq\) A i.e. B is proper subset of A**Improper Subset**Eg: A = {2, 4, 6, 8, 10}, B = {2, 4, 8} and C = {10, 8, 6, 2, 4}

C \(\subseteq\) A \(\Rightarrow\) C is improper subset of A

**Universal Sets **A set containing all the sets under consideration or (Supposition of sets) is called universal set. It is denoted by U.

U = {1, 2, 3, 4, 5 . . . . .}

U = {set of natural numbers}

**NOTE:**All the other sets will be universal set under consideration.

Eg:

- A = {boys students of a class}

B = {girls students of a class}

Universal set for both A and B will be U = {students of a class} - U = {1, 2, 3, 4, . . . . . . . }

A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}

A \(\subset\) U and B \(\subset\) U

So, Universal set is a set which contains all other sets under consideration as subsets.

**Number of Subsets **Number of subsets depends on the number of elements present in the set. If a set A contains 'n' elements, the number of subsets of the set A are 2

Eg:

A = {2, 4}

Subset of A are {2}, {4}, {2, 4} and { }.

\(\therefore\) Number of elements n(A) = 2

Number of subsets = 2

**NOTE:**Every set is subset of itself

Null set / empty set is subset of every set.

**Examples:**

**Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Write down the elements of the set B if B = {x : x \(\in\) A}**Solution:

Here,

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

A = {1, 3, 5, 7, 9}

then,

B \(\subset\) U

B = {2, 4, 6, 8, 10} the elements which does not belong to A.**If U = N and A = {x : x < 4}, list the elements of set A. Also find n(A)**Solution:

The Universal set (U) = N = {1, 2, 3, 4 . . . . . }

\(\therefore\) A = {x : x < 4} = {1, 2, 3} and n(A) = 3**State the set relation for the following pair sets (Overlapping or disjoint)**

**A = {factors of 12}, B = {factor of 18}**Solution:

A = {factors of 12}

= {1, 2, 3, 4, 6, 12}

B = {factors of 18}

= {1, 2, 3, 9, 18}

A and B are Overlapping sets**A = {2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 9}**Solution:A = {2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 9}

A and B are disjoint sets

**State whether thee following pairs of sets are equal or equivalent****A = {multiple of 2 less than 12}, B = {even numbers less than 12}**Solution:

A = {2, 4, 6, 8, 10} , B = {2, 4, 6, 8, 10}

A and B are equal sets**A = {Vowels of English alphabets}, B = {first five letters of English alphabets}**Solution:

A = {a, e, i, o, u}, B = {a, b, c, d, e}

n(A) = 5, n(B) = 5

A and B are equivalent sets

**State Universal set and its subsets from the following pair of sets.**

**P = {students of class 7}, Q = {students of school}**Solution:

P = {subset of Q i.e. P \(\subset\) Q}

Q is universal set of P**A = {Domestic animals}, B = {cow, goat, buffalo}**Solution:

B is subset of A, i.e. B \(\subset\) A and A is universal set of B.

**Exercise **

- Make suitable Universal set for
- A = {Natural Number}
- B = {factors of 6}
- P = {Animals in a house}
- Q = {Domestic Animals}

- Make possible subsets from the following sets
- {a}
- {a, e}
- {1, 2, 3}

- List the common element from the pair of sets given:
- A = {x : x is multiple of 4, x \(\leq\) 24} and B = {x : x is multiple of 6, x \(\leq\) 24}
- P = {factors of 6} and Q = {factors of 8}

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

- Union of sets
- Intersection of sets
- Difference of sets Complement of sets

**Union of Sets **When the elements of two or more than two sets are listed together in a single set is known as Union of sets.

The Union of two sets A and B is the set containing all the elements present either in sets A or B or both A and B. It is denoted by A \(\cup\) B.

Eg:

A = {2, 3, 5, 7}, B = {2, 4, 6, 8}

A \(\cup\) B = {2, 3, 5, 7} \(\cup\) {2, 4, 6, 8}

A \(\cup\) B = {2, 3, 4, 5, 6, 7, 8}

Here, 2 is repeated but written only once.

**NOTE: **Repeated elements are written for single time.

**Intersection of Sets **When the common elements present in two or more than two sets are listed together in separate set is known as intersection of sets.

The intersection of two sets A and B is the set containing all the common elements present both sets A and B. It is denoted by (A \(\cap\) B)

Eg:

A = {factors of 12}, B = {factors of 18}

A = {1, 2, 3, 4, 6, 12}, B = {1, 2, 3, 6, 9, 18}

A \(\cap\) B = {1, 2, 3, 4, 6, 12} \(\cap\) {1, 2, 3, 6, 9, 18}

= {1, 2, 3, 6}

**Difference of Sets **When the elements present only in 1

The difference of sets A and B is the set if all elements present only in set A but not in set B. It is denoted by A – B.

Eg:

A = {1, 2, 3, 4, 6, 12}, B = {1, 2, 3, 6, 9, 18}

A – B = {1, 2, 3, 4, 6, 12} – {1, 2, 3, 6, 9, 18}

= {4, 12}

i.e. the element present only in A set is {4, 12}

Similarly,

B – A = {1, 2, 3, 6, 9, 18} – {1, 2, 3, 4, 6, 12}

= {9, 18}

= Elements present only in set B.

**Complement of Sets **When the elements present only in Universal set except the elements of given set written in a separate set is known as complement of set.

The complement of set A is the elements of Universal set but not the elements of set A are the complement of set A. It is denoted by letter with bar or dash.

Eg:

If A = {1, 3, 5, 7, 9} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

\(\bar{A}\) = U – A

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

= {2, 4, 6, 8, 10}

**Examples**

**Let U = {1, 2, 3, 4, 5, 6, 7,8, 9, 10} , A = {3, 4, 5, 6, 7} and B = {2, 4, 6, 8, 10}. Find, **

**A \(\cup\) B****A \(\cap\) B****\(\overline{A \cup B}\)****\(\overline{A \cap B}\)****\(\overline{A}\)****\(\overline{B}\)****\(\overline{A}\) \(\cap\) \(\overline{B}\)****A – B****B – A****\(\overline{A – B}\)****\(\overline{A}\) \(\cup\) (A – B)****\(\overline{A}\) \(\cap\) \(\overline{B}\)**

Solution:

Here U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {3, 4, 5, 6, 7} and B = {2, 4 , 6, 8, 10}

- A \(\cup\) B = {3, 4, 5, 6, 7} \(\cup\) {2, 3, 4, 6, 8, 10}

= {2, 3 ,4 5, 6, 7, 8, 10} - A \(\cap\) B = {3, 4, 5, 6, 7} \(\cap\) {2, 4, 6, 8, 10}

= {4, 6} - \(\overline{A \cup B}\) = U – (A \(\cup\) B)

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – [{3, 4, 5, 6, 7} \(\cup\) {2, 4, 6, 8, 10}]

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

= {1, 9} - \(\overline{A \cap B}\) = U – {A \(\cap\) B}

A \(\cap\) B = {3, 4, 5, 6, 7} \(\cap\) {2, 4, 6, 8, 10}

= {4, 6}

\(\overline{A \cap B}\) = U – {A \(\cap\) B}

= {1, 2, 3, . . . 10} – {4, 6}

= {1, 2, 3, 5, 7, 8, 9, 10} - \(\overline{A}\) = U – A

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

= {1, 2, 8, 9, 10} - \(\overline{B}\) = U – B

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

= {1, 3, 5, 7, 9} - \(\overline{A}\) \(\cup\) \(\overline{B}\)

\(\overline{A}\) = {1, 2, 8, 9, 10} , \(\overline{B}\) = {1, 3, 5, 7, 9}

\(\overline{A}\) \(\cup\) \(\overline{B}\) = {1, 2, 8, 9, 10} \(\cup\) {1, 3, 5, 7, 9}

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

i.e. \(\overline{A} \) \(\cup\) \(\overline{B}\) = \(\overline{A \cap B}\) - A – B = {3, 4, 5, 6, 7} – {2, 4, 6, 8, 10}

= {3, 5, 7} - B – A = {2, 4, 6, 8, 10} – {3, 4, 5, 6, 7}

= {2, 8 , 10} - \(\overline{A – B}\)

A – B = {3, 4, 5, 6, 7} – {2, 4, 6, 8, 10}

= {3, 5, 7}

\(\overline{A – B}\) = U – (A – B)

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

= {1, 2, 4, 6, 8, 9, 10} - \(\overline{B – A}\)

B – A = {2, 4, 6, 8, 10} – {3, 4, 5, 6, 7}

= {2, 8, 10}

\(\overline{B – A}\) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {2, 8, 10}

= {1, 3, 4, 5, 6, 7, 9} - \(\overline{A}\) \(\cap\) \(\overline{B}\)

\(\overline{A}\) = {1, 2, 8, 9, 10}

\(\overline{B}\) = {1, 3, 5, 7, 9}

\(\overline{A}\) \(\cap\) \(\overline{B}\) = {1, 2, 8, 9, 10} \(\cap\) {1, 3, 5, 7, 9}

= {1, 9}

\(\overline{A}\) \(\cap\) = \(\overline{A \cup B}\)

**Try these **

- If U = {1, 2, 3 . . . . . 20}, A = {1, 3, 5, 7, 11, 13, 15}, B = {3, 6, 9, 12, 15, 18} and C = {1, 2, 3, 4, ,5 6, 7, 8}. Find

(i) A \(\cup\) B

(ii) A \(\cap\) B

(iii) A – B

(iv) A \(\cup\) B \(\cup\) C

(v) A \(\cap\) B \(\cap\) C

(vi) A – (B \(\cup\) C)

(vii) \(\overline{A \cup B \cup C}\)

(viii) \(\overline{A – B}\)

(ix) \(\overline{B – C}\)

(x) \(\overline{A \cap B}\) - From the venn-diagram, list the elements of following sets
- A \(\cup\) B
- (\(\overline{A \cap B}\))
- (A – B)
- \(\overline{A \cup B}\)
- \(\overline{A – B}\) \(\overline{B – A}\)

- If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {3, 4, 5, 6, 7}, B = {2, 4 ,6, 8, 10} and C = {2, ,3 5 ,9}. Find the elements of following sets.
- A \(\cup\) B \(\cup\) C
- \(\overline{A \cup B \cup C}\)
- A \(\cap\) B \(\cap\) C
- \(\overline{A \cap B \cap C}\)
- \(\overline{A}\) \(\cup\) \(\overline{B}\) \(\cup\) \(\overline{C}\)
- \(\overline{A}\) \(\cap\) \(\overline{B}\) \(\cap\) \(\overline{C}\)

- Name the shaded region in the venn – diagram;
- If A = {a, e, I, o, u} and B = {a, b, c, d, e, f}, find A \(\cup\) B, A \(\cap\) B, A – B and B – A.
- If A = {natural number less than 9}, B = {prime number less than 15}, find A \(\cup\) B, A \(\cap\) B, A – B and B – A.
- If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}, find A \(\cap\) B, \(\overline{A}\), \(\overline{B}\) and \(\overline{A \cup B}\).
- If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {odd numbers} and B = {prime numbers}, find \(\overline{A \cap B}\), \(\overline{B}\), A \(\cap\) B and \(\overline{A – B}\)
- If C = {A, b, c, d, e, f, g, h, i, j}, A = {a, e, i, o, u} and B = {a, b, c, d, e}, then prove:
- A \(\cup\) B = B \(\cup\) A
- A \(\cap\) B = B \(\cap\) A
- A \(\cap\) B \(\cap\) C = C \(\cap\) B \(\cap\) A
- \(\overline{A \cap B \cap C}\) = \(\overline{A \cap B}\) \(\cup\) \(\overline{B \cap C}\)

© 2019 EDUKUM.COM ALL RIGHTS RESERVED