Sets - Notes, Concept and All Important Formula

 SETS

1. SET :

A set is a collection of well defined objects which are distinct from each other.

Set are generally denoted by capital letters $A, B, C, \ldots .$ etc. and the elements of the set by a, b, c .... etc.

If a is an element of a set $A$, then we write $a \in A$ and say a belongs to $A$.

If a does not belong to $A$ then we write $a \notin A$,

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All Chapter Notes, Concept and Important Formula

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2. SOME IMPORTANT NUMBER SETS :

$N =$ Set of all natural numbers

$=\{1,2,3,4, \ldots\}$

$W =$ Set of all whole numbers

$=\{0,1,2,3, \ldots .\}$

$Z$ or I set of all integers

$\{\ldots-3,-2,-1,0,1,2,3, \ldots\}$

$Z ^{+}$= Set of all +ve integers 

$=\{1,2,3, \ldots\}= N .$

$Z ^{-}$ = Set of all -ve integers

$=(-1,-2,-3, \ldots .\}$

$Z _{0}=$ The set of all non-zero integers.

$=\{\pm 1, \pm 2, \pm 3, \ldots\}$

Q = The set of all rational numbers. 

$=\left\{\frac{p}{q}: p, q \in I, q \neq 0\right\}$

$R$= the set of all real numbers. 

$R - Q =$ The set of all irrational numbers

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3. REPRESENTATION OF A SET :

(i) Roster Form : In this form a set is described by listing elements, separated by commas and enclose then by curly brackets

(ii) Set Builder Form : In this case we write down a property or rule $p$ Which gives us all the element of the set $A=\{x: P(x)\}$

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4. TYPES OF SETS :

Null set or Empty set : A set having no element in it is called an Empty set or a null set or void set it is denoted by $\phi$ or { } A set consisting of at least one element is called a non-empty set or a non-void set.

Singleton : A set consisting of a single element is called a singleton set.

Finite Set : A set which has only finite number of elements is called a finite set.

Order of a finite set : The number of elements in a finite set is called the order of the set $A$ and is denoted $O ( A )$ or $n ( A )$. It is also called cardinal number of the set.

Infinite set : A set which has an infinite number of elements is called an infinite set.

Equal sets : Two sets $A$ and $B$ are said to be equal if every element of $A$ is a member of $B$, and every element of $B$ is a member of $A$. If sets $A$ and $B$ are equal. We write $A=B$ and if $A$ and $B$ are not equal then $A \neq B$.

Equivalent sets : Two finite sets $A$ and $B$ are equivalent if their number of elements are same i.e. $n ( A )= n ( B )$

Note : Equal sets are always equivalent but equivalent sets may not be equal.

Subsets : Let $A$ and $B$ be two sets if every element of $A$ is an element B, then $A$ is called a subset of $B$ i.e. $A \subseteq B$

Proper subset : If $A$ is a subset of $B$ and $A \neq B$ then $A$ is a proper subset of $B$. and we write $A \subset B$

Note-1 : Every set is a subset of itself i.e. $A \subseteq A$ for all $A$

Note-2 : Empty set $\phi$ is a subset of every set 

Note-3 : Clearly $N \subset W \subset Z \subset Q \subset R \subset C$

Note- 4 : The total number of subsets of a finite set containing $n$ elements is $2^{n}$

Universal set : A set consisting of all possible elements which occur in the discussion is called a Universal set and is denoted by $U$

Note : All sets are contained in the universal set 

Power set : Let A be any set. The set of all subsets of A is called power set of $A$ and is denoted by $P(A)$

Some Operation on Sets:

(i) Union of two sets : $A \cup B=\{x: x \in A$ or $x \in B\}$

(ii) Intersection of two sets : $A \cap B=\{x: x \in A$ and $x \in B\}$

(iii) Difference of two sets : $A-B=\{x: x \in A$ and $x \notin B\}$

(iv) Complement of a set : $A^{\prime}=\{x: x \notin A$ but $x \in U\}=U-A$

(v) De-Morgan Laws : $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} ;(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$

(vi) $A-(B \cup C)=(A-B) \cap(A-C) ;$$A-(B \cap C)=(A-B) \cup(A-C)$

(vii) Distributive Laws : $A \cup(B \cap C)=(A \cup B) \cap(A \cup C) ; A \cap(B \cup C)$$=(A \cap B) \cup(A \cap C)$

(viii) Commutative Laws : $A \cup B=B \cup A ; A \cap B=B \cap A$

(ix) Associative Laws : $(A \cup B) \cup C=A \cup(B \cup C) ;(A \cap B) \cap C$$=A \cap(B \cap C)$

(x) $A \cap \phi=\phi ; A \cap U=A$

$A \cup \phi=A ; A \cup U=U$

(xi) $A \cap B \subseteq A ; A \cap B \subseteq B$

(xii) $A \subseteq A \cup B ; B \subseteq A \cup B$

(xiii) $A \subseteq B \Rightarrow A \cap B=A$

(xiv) $A \subseteq B \Rightarrow A \cup B=B$

Disjoint Sets :

IF $A \cap B=\phi$, then $A, B$ are disjoint. Note : $A \cap A ^{\prime}=\phi \quad \therefore A , A ^{\prime}$ are disjoint.

Symmetric Difference of Sets:

$A \Delta B=(A-B) \cup(B-A)$

• $\left( A ^{\prime}\right)^{\prime}= A$
•  $\quad A \subseteq B \Leftrightarrow B^{\prime} \subseteq A^{\prime}$

If $A$ and $B$ are any two sets, then

(i) $\quad A-B=A \cap B^{\prime}$

(ii) $\quad B-A=B \cap A^{\prime}$

(iii) $\quad A-B=A \Leftrightarrow A \cap B=\phi$

(iv) $\quad(A-B) \cup B=A \cup B$

(v) $\quad(A-B) \cap B=\phi$

(vi) $\quad(A-B) \cup(B-A)=(A \cup B)-(A \cap B)$

Venn Diagram :

Note : $A \cap A^{\prime}=\phi, A \cup A^{\prime}=U$

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5. SOME IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS :

If $A, B$ and $C$ are finite sets, and $U$ be the finite universal set, then

(i) $n ( A \cup B )= n ( A )+ n ( B )- n ( A \cap B )$

(ii) $n ( A \cup B )= n ( A )+ n ( B ) \Leftrightarrow A , B$ are disjoint sets

(iii) $n(A-B)=n(A)-n(A \cap B) \text { i.e. }$$n(A-B)+n(A \cap B)=n(A)$

(iv) $n ( A \Delta B )=$ No. of elements which belong to exactly one of $A$ or $B$

= $n (( A - B ) \cup( B - A ))$

= $n ( A - B )+ n ( B - A )[\because( A - B )$ and $( B - A )$ are disjoint $]$

=$n ( A )- n ( A \cap B )+ n ( B )- n ( A \cap B )$

=$n ( A )+ n ( B )-2 n ( A \cap B )$

= $n ( A )+ n ( B )-2 n ( A \cap B )$

(v) $n ( A \cup B \cup C )$$= n ( A )+ n ( B )+$$n ( C )-$$n ( A \cap B )-$$n ( B \cap C )-$$n ( A \cap C )+$$n ( A \cap B \cap C )$

(vi) Number of elements in exactly two of the sets $A, B, C$

= $n ( A \cap B )+$$n ( B \cap C )+$$n ( C \cap A )-$$3 n ( A \cap B \cap C )$

(vii) number of elements in exactly one of the sets $A , B , C$

=$n(A)+n(B)+n(C)-$$2 n(A \cap B)-$$2 n(B \cap C)-$$2 n(A \cap C)$$+3 n(A \cap B \cap C)$

(viii) $n\left(A^{\prime} \cup B^{\prime}\right)=n\left((A \cap B)^{\prime}\right)=n(U)-n(A \cap B)$ 

(ix) $n\left(A^{\prime} \cap B^{\prime}\right)=n\left((A \cup B)^{\prime}\right)=n(U)-n(A \cup B)$

(x) If $A_{1}, A_{2} \ldots \ldots . A_{n}$ are finite sets, then

$n \left(\bigcup_{ i =1}^{ n } A _{ i }\right)=\displaystyle\sum_{ i =1}^{ n } n \left( A _{ i }\right)-\sum_{1 \leq i < j \leq n } n \left( A _{ i } \cap A _{ j }\right)$

$+\displaystyle \sum_{1 \leq i \lt j \lt k \leq n } n \left( A _{ i } \cap A _{ j } \cap A _{ k }\right)-\ldots$$\ldots+(-1)^{ n -1} n \left( A _{1} \cap A _{2} \cap \ldots . A _{ n }\right)$

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