A Set is a Collection of Distinct Items of Similar or Dis-Similar Types.
Items in a Set are called Elements. Any Element can only be present Once in a Set. Element Duplication is Not Allowed.
A Set with Zero Elements is called a Null Set or an Empty Set. It is denoted by the Greek Letter \(\varphi\) or a Pair of Empty Braces \(\{\}\).
A Non Empty Set with One or More Elements is generally denoted by a Capital English or Greek Alphabet.
The Elements of the Set can be represented by any of the following methods
Explicitly writing the Elements between a Pair of Braces Separated by Comma as follows
\(A=\{B,E,G,Q\}\) (Set of 4 English Capital Letter Alphabets) ...(1)
\(B=\{17,12,19,8,3,9\}\) (Set of 6 Decimal Numbers) ...(2)
\(C=\{B,12,G,8,L,5,7,P\}\) (Set of Some Decimal Numbers and Capital Alphabets) ...(3)
Specifying the Type of Set and/or One or More Conditions Between or Without Pair of Braces Without Using Variables as follows
\(E=\{Set\hspace{.1cm}of\hspace{.1cm}All\hspace{.1cm}Integers\hspace{.1cm}\leq\hspace{.1cm}5\}\) (Set Represented by Specifying Some Condition) ...(4)
\(F=Set\hspace{.1cm}of\hspace{.1cm}All\hspace{.1cm}Real\hspace{.1cm}Numbers\) (Set Represented by Specifying Type of Set) ...(5)
Specifying the Type of Set and/or One or More Conditions Between Pair of Braces Using Variables as follows
\(G=\{x : x>5, x <20\}\) (Set Represented by using Variable \(x\), such that \(x\) is a number Greater than 5 and Lesser than 20 ) ...(6)
\(H=\{y\hspace{.2cm}|\hspace{.2cm}y>0, y \leq 50\}\) (Set Represented by using Variable \(y\), such that \(y\) is a number Greater than 0 and Lesser than or Equal to 50) ...(7)
Please Note that the Order of Elements in a Set Does Not Matter. That is, Any 2 or more Sets are Same as long as the Elements of those Sets are Same.
Sets containing only a Single Element are called Singleton Sets or Singletons.
If an Element is Present in any given Set, it is said to Belong to that Set. It is denoted by the symbol \(\in\).
For example, if an Element \(x\) is Present in Set \(A\) it is represented as \(x \in A\).
If an Element is Not Present in any given Set, it is said that it Does Not Belong to that Set. It is denoted by the symbol \(\notin\).
For example, if an Element \(x\) is Not Present in Set \(A\) it is represented as \(x \notin A\).
Any given Set \(A\) it is said to be a Proper Subset of another Set \(B\) if
Set \(B\) has all Elements of Set \(A\) along with some additional Elements. Set \(A\) is said to be a Subset of Set \(B\) if either Set \(A\) is a Proper Subset or Equal to Set \(B\).
The following example demonstrates how Set-Subset Relations are represented
\(A \subset B\) (Represents Set \(A\) is a Proper Subset of Set \(B\)) ...(8)
\(A \not\subset B\) (Represents Set \(A\) is Not a Proper Subset of Set \(B\)) ...(9)
\(A \subseteq B\) (Represents Set \(A\) is Either a Proper Subset of or Equal to Set \(B\). Also said as Set \(A\) is a Subset of Set \(B\))) ...(10)
\(A \nsubseteq B\) (Represents Set \(A\) is Niether a Proper Subset of Nor Equal to Set \(B\). Also said as Set \(A\) is Not a Subset of Set \(B\)) ...(11)
Please Note that the Null/Empty Set \(\varphi\) is a Subset to Every Set.
Any given Set \(A\) it is said to be a Proper Superset of another Set \(B\) if
Set \(A\) has all Elements of Set \(B\) along with some additional Elements. Set \(A\) is said to be a Superset of Set \(B\) if either Set \(A\) is a Proper Suuperset or Equal to Set \(B\).
The following example demonstrates how Set-Superset Relations are represented
\(A \supset B\) (Represents Set \(A\) is a Proper Superset of Set \(B\)) ...(12)
\(A \not\supset B\) (Represents Set \(A\) is Not a Proper Superset of Set \(B\)) ...(13)
\(A \supseteq B\) (Represents Set \(A\) is Either a Proper Superset of or Equal to Set \(B\). Also said as Set \(A\) is a Superset of Set \(B\)) ...(14)
\(A \nsupseteq B\) (Represents Set \(A\) is Niether a Proper Superset of Nor Equal to Set \(B\). Also said as Set \(A\) is Not a Superset of Set \(B\)) ...(15)
A Set containing Countable (i.e. Finite Number) of Elements is called a Finite Set. A Set containing Uncountable (i.e. Infinite Number) of Elements is called an Infinite Set.
For example A Set of All English Alphabets is a Finite Set, whereas A Set of All Integers is an Infinite Set.
The Number of Elements present in a Finite Set is called the Cardinality of the Set.
Given any Set \(A\), the Cardinality of the Set is denoted by \(|A|\).
The Elements of 2 or more Sets can be combined together into a Common Set through the Operations of Union or Intersection.
The Operation of Union between 2 Sets is represented and performed by the Operator \(\cup\). The Union of 2 or more Sets gives a Resultant Set that has all the Elements of Participating Sets with the Elements that are Common to 2 or more Sets represented only once (i.e. the Duplicates are removed).
The following gives the Result of Union Operation for the Sets \(A\), \(B\) and \(C\) as defined in equations (1), (2) and (3) above
\(A \cup B =\{B,E,G,Q\} \cup \{17,12,19,8,3,9\}=\{B,E,G,Q,17,12,19,8,3,9\}\) ...(16)
\(A \cup C =\{B,E,G,Q\} \cup \{B,12,G,8,L,5,7,P\}=\{B,E,G,Q,12,8,L,5,7,P\}\) ...(17)
\(B \cup C =\{17,12,19,8,3,9\} \cup \{B,12,G,8,L,5,7,P\}=\{17,12,19,8,3,9,B,G,L,5,7,P\}\) ...(18)
\(A \cup B \cup C =\{B,E,G,Q\} \cup \{17,12,19,8,3,9\} \cup \{B,12,G,8,L,5,7,P\}=\{B,E,G,Q,17,12,19,8,3,9,L,5,7,P\}\) ...(19)
The Resultant Set of the Union Operation on all Available Sets is called the Universal Set. The Universal Set is denoted by letter \(U\).
As per the above example
\(U=A \cup B \cup C=\{B,E,G,Q,17,12,19,8,3,9,L,5,7,P\}\) ...(20)
Union of \(N\) number of Sets \(A_1, A_2, A_3, ..., A_N\) is denoted as follows
Please Note that Any Set formed as a result of Union of Other Sets is a Superset of all the Participating Sets. Hence, the Universal Set
is the Superset of All Availble Sets (i.e. All Availble Sets are Subsets of the Universal Set).
The Operation of Intersection between 2 Sets is represented and performed by the Operator \(\cap\). The Intersection of 2 or more Sets gives a Resultant Set that has only the Elements Common to all the Participating Sets. If the Participating Sets do not have any Element in Common the Resultant Set is a Null Set.
The following gives the Result of Intersection Operation for the Sets \(A\), \(B\) and \(C\) as defined in equations (1), (2) and (3) above
\(A \cap B =\{B,E,G,Q\} \cap \{17,12,19,8,3,9\}=\{\}=\varphi\) ...(22)
\(A \cap C =\{B,E,G,Q\} \cap \{B,12,G,8,L,5,7,P\}=\{B,G\}\) ...(23)
\(B \cap C =\{17,12,19,8,3,9\} \cap \{B,12,G,8,L,5,7,P\}=\{12,8\}\) ...(24)
\(A \cap B \cap C =\{B,E,G,Q\} \cap \{17,12,19,8,3,9\} \cap \{B,12,G,8,L,5,7,P\}=\{\}=\varphi\) ...(25)
Intersection of \(N\) number of Sets \(A_1, A_2, A_3, ..., A_N\) is denoted as follows
Please Note that Any Set formed as a result of Intersection of Other Sets is a Subset of all the Participating Sets.
Also 2 or more Sets having No Elements in Common are called Disjoint Sets. The Intersection of Disjoint Sets is the Null/Empty Set \(\varphi\).
Any given Set \(A\) can be Subtracted from another given Set \(B\) only if Set \(A\) is a Subset of Set \(B\).
This Difference Operation gives a Resultant Set \(D\) as follows
\(D=B-A\) (Only if \(A \subseteq B\)) ...(27)
This Set \(D\) Contains all Elements of Set \(B\) that Do Not Also Belong to Set \(A\). Therefore, if \(B=A\) then \(D\) is a Null Set. That is
\(D=B-A=\{\}=\varphi\) (Only if \(B=A\)) ...(28)
The Complement / Inverse of any given Set \(A\) (denoted by \(\overline{A}\)) Contains All Elements that Do Not Belong to Set \(A\).
Since All Availble Sets are Subsets of the Universal Set, the Complement / Inverse of any Set can be found out by Subtracting it from the Universal Set. Hence, Set \(\overline{A}\) is calculated as follows
\(\overline{A}= U-A\) ...(29)
The following calculates the Complement / Inverse of Sets \(A\), \(B\) and \(C\) as defined in equations (1), (2) and (3) above by Subtracting them from Universal Set as given in equation (20) above
The Complements of Union and Intersection of \(N\) number of Sets \(A_1, A_2, A_3, ..., A_N\) are denoted as follows
\(\overline{\bigcup \limits_{i=1}^N A_i}=\overline{A_1 \cup A_2 \cup A_3 \cup \cdots \cup A_N}\) (Complement of Union of \(N\) number of Sets) ...(33)
\(\overline{\bigcap \limits_{i=1}^N A_i}=\overline{A_1 \cap A_2 \cap A_3 \cap \cdots \cap A_N}\) (Complement of Intersection of \(N\) number of Sets) ...(34)
Given any Set \(S\), the Powerset \(P(S)\) is a Set that contains as its Elements All Subsets of the Set \(S\) (including the Null/Empty Set and the Set itself).
If the Cardinality of the Set \(S\) is \(N\), Cardinality of the Corresponding Powerset \(P(S)\) is \(2^N\).