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Set

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

1. 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
2. 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
3. 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:

1. 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
2. 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.
3. 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}
4. 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:

1. Proper Subset
Eg: A = {1, 2, 3, 4, 5}, B = {2, 4, 5}
B $\subseteq$ A i.e. B is proper subset of A
2. 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 2n.
Eg:
A = {2, 4}
Subset of A are {2}, {4}, {2, 4} and { }.
$\therefore$ Number of elements n(A) = 2
Number of subsets = 2n = 22 = 4

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 1st set is written in separate set is known as difference of two sets.
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,

1. A $\cup$ B
2. A $\cap$ B
3. $\overline{A \cup B}$
4. $\overline{A \cap B}$
5. $\overline{A}$
6. $\overline{B}$
7. $\overline{A}$ $\cap$ $\overline{B}$
8. A – B
9. B – A
10. $\overline{A – B}$
11. $\overline{A}$ $\cup$ (A – B)
12. $\overline{A}$ $\cap$ $\overline{B}$

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}

1. A $\cup$ B = {3, 4, 5, 6, 7} $\cup$ {2, 3, 4, 6, 8, 10}
= {2, 3 ,4 5, 6, 7, 8, 10}
2. A $\cap$ B = {3, 4, 5, 6, 7} $\cap$ {2, 4, 6, 8, 10}
= {4, 6}
3. $\overline{A \cup B}$ = 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}
4. $\overline{A \cap B}$ = 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}
5. $\overline{A}$ = U – A
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {3, 4, 5, 6, 7}
= {1, 2, 8, 9, 10}
6. $\overline{B}$ = U – B
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {2, 4, 6, 8, 10}
= {1, 3, 5, 7, 9}
7. $\overline{A}$ $\cup$ $\overline{B}$
$\overline{A}$ = {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. $\overline{A}$ $\cup$ $\overline{B}$ = $\overline{A \cap B}$
8. A – B = {3, 4, 5, 6, 7} – {2, 4, 6, 8, 10}
= {3, 5, 7}
9. B – A = {2, 4, 6, 8, 10} – {3, 4, 5, 6, 7}
= {2, 8 , 10}
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}
11. $\overline{B – A}$
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}
12. $\overline{A}$ $\cap$ $\overline{B}$
$\overline{A}$ = {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;  (I) (II) (III) (IV)
• 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
• $\overline{A \cap B \cap C}$ = $\overline{A \cap B}$ $\cup$ $\overline{B \cap C}$