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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 \(a \notin A\),


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


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 \(A=\{x: P(x)\}\)


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 :

Venn diagram sets

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


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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