Sets

$\displaystyle \textbf{What is a Set?}$

$\displaystyle \text{A set is a well-defined collection of distinct objects.}$
$\displaystyle \text{Example: }A=\{1,2,3,4,5\}.$

$\displaystyle \textbf{What is an Element of a Set?}$

$\displaystyle \text{The objects in a set are called its elements.}$

$\displaystyle \text{For example, in the set }A=\{1,2,3,4,5\},\text{ the elements are }1,2,3,4\text{ and }5.$
$\displaystyle \text{Thus, we can write }1\in A,\ 2\in A,\ 3\in A,\ 4\in A\text{ and }5\in A.$

$\displaystyle \text{Usually, sets are denoted by capital letters such as }A,B,C,\text{ etc., while their}$
$\displaystyle \text{elements are generally denoted by small letters such as }x,y,z,\text{ etc.}$

$\displaystyle \text{If }x\text{ is an element of }A,\text{ then we say that }x\text{ belongs to }A\text{ and write }x\in A.$

$\displaystyle \text{If }x\text{ is not an element of }A,\text{ then we say that }x\text{ does not belong to }A$
$\displaystyle \text{and write }x\notin A.$

$\displaystyle \\$

$\displaystyle \textbf{How to Describe a Set?}$

$\displaystyle \textbf{1. Roster Method or Tabular Form}$

$\displaystyle \text{In this form, we list all the elements of the set within braces.}$
$\displaystyle \text{Example: }A=\{1,2,3,4\}\text{ or }B=\{a,b,c,d,e\}.$

$\displaystyle \textbf{2. Set-Builder Form or Rule Method or Description Method}$

$\displaystyle \text{In this form, we describe the property or rule satisfied by all the elements of the set.}$
$\displaystyle \text{Example: }A=\{x:x\in N,\ x<5\}.$

$\displaystyle \textbf{Some Examples of Roster Form vs Set-Builder Form}$

$\displaystyle \begin{array}{|c|c|c|} \hline \text{S. No.}&\text{Roster Form}&\text{Set-Builder Form}\\ \hline 1&\{1,2,3,4,5\}&\{x:x\in N,\ x<6\}\\ \hline 2&\{2,4,6,8,10\}&\{x:x=2n,\ n\in N,\ 1\leq n\leq5\}\\ \hline 3&\{1,4,9,16,25,36\}&\{x:x=n^2,\ n\in N,\ 1\leq n\leq6\}\\ \hline \end{array}$

$\displaystyle \textbf{Sets of Numbers}$

$\displaystyle \textbf{Natural Numbers }(N)$
$\displaystyle N=\{1,2,3,4,5,6,7,\ldots\}.$

$\displaystyle \textbf{Whole Numbers }(W)$
$\displaystyle W=\{0,1,2,3,4,5,6,\ldots\}.$

$\displaystyle \textbf{Integers }(Z)$
$\displaystyle Z=\{\ldots,-3,-2,-1,0,1,2,3,4,\ldots\}.$

$\displaystyle \textbf{Rational Numbers }(Q)$
$\displaystyle Q=\left\{\frac{p}{q}:p\in Z,\ q\in Z,\ q\neq0\right\}.$

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$\displaystyle \textbf{Finite Sets and Infinite Sets}$

$\displaystyle \textbf{Finite Set}$

$\displaystyle \text{A set having a finite number of distinct elements is called a finite set.}$
$\displaystyle \text{In other words, the process of counting its elements comes to an end.}$

$\displaystyle \text{Examples:}$
$\displaystyle \text{(i) The set of all natural numbers less than }50.$
$\displaystyle \text{(ii) The set of all factors of }36.$

$\displaystyle \textbf{Infinite Set}$

$\displaystyle \text{A set having infinitely many distinct elements is called an infinite set.}$

$\displaystyle \text{Example: The set of all natural numbers greater than }100.$
$\displaystyle \{x:x\in N,\ x>100\}=\{101,102,103,\ldots\}.$

$\displaystyle \textbf{Cardinal Number of a Finite Set}$

$\displaystyle \text{The number of distinct elements contained in a finite set }A\text{ is called the cardinal}$
$\displaystyle \text{number of }A\text{ and is denoted by }n(A).$

$\displaystyle \text{Example 1: }A=\{1,2,3,4\}.\text{ Therefore, }n(A)=4.$

$\displaystyle \text{Example 2: }A=\{x:x\text{ is a letter in the word 'APPLE'}\}.$
$\displaystyle \text{Since repeated elements are written only once in a set, }A=\{A,P,L,E\}.$
$\displaystyle \therefore n(A)=4.$

$\displaystyle \text{Example 3: }A=\{x:x\text{ is a factor of }36\}.$
$\displaystyle A=\{1,2,3,4,6,9,12,18,36\}.$
$\displaystyle \therefore n(A)=9.$

$\displaystyle \textbf{Empty Set}$

$\displaystyle \text{A set containing no elements is called an empty set, a null set or a void set.}$
$\displaystyle \text{It is denoted by }\phi.$
$\displaystyle \text{In roster form, }\phi=\{\}.$
$\displaystyle \text{Also, }n(\phi)=0.$

$\displaystyle \text{Examples:}$
$\displaystyle \text{(i) }\{x:x\in N,\ 3<x<4\}=\phi.$
$\displaystyle \text{(ii) }\{x:x\text{ is an even prime number and }x>5\}=\phi.$

$\displaystyle \textbf{Non-Empty Set}$

$\displaystyle \text{A set which has at least one element is called a non-empty set.}$
$\displaystyle \text{Example: }A=\{1,2,3\}\text{ or }B=\{1\}.$

$\displaystyle \textbf{Singleton Set}$

$\displaystyle \text{A set containing exactly one element is called a singleton set.}$
$\displaystyle \text{Example: }A=\{a\}\text{ or }B=\{1\}.$

$\displaystyle \\$

$\displaystyle \textbf{Equal Sets}$

$\displaystyle \text{Two sets }A\text{ and }B\text{ are said to be equal sets, written as }A=B,\text{ if every}$
$\displaystyle \text{element of }A\text{ is in }B\text{ and every element of }B\text{ is in }A.$

$\displaystyle \text{Example: }A=\{1,2,3,4\}\text{ and }B=\{4,2,3,1\}.\text{ Therefore, }A=B.$

$\displaystyle \text{Equality of sets depends on the elements themselves, not merely on the number of elements.}$

$\displaystyle \text{If the sets are not equal, then we write }A\neq B.$

$\displaystyle \textbf{Equivalent Sets}$

$\displaystyle \text{Two finite sets }A\text{ and }B\text{ are said to be equivalent if }n(A)=n(B),$
$\displaystyle \text{i.e. they have the same number of elements.}$

$\displaystyle \text{Example: }A=\{a,e,i,o,u\}\text{ and }B=\{1,2,3,4,5\}.$
$\displaystyle n(A)=5\text{ and }n(B)=5.\text{ Therefore, }A\text{ and }B\text{ are equivalent sets.}$

$\displaystyle \text{Note: Two equal sets are always equivalent, but two equivalent sets need not be equal.}$

$\displaystyle \textbf{Subsets}$

$\displaystyle \text{If }A\text{ and }B\text{ are two sets such that every element of }A\text{ is also an element of }B,$
$\displaystyle \text{then }A\text{ is called a subset of }B,\text{ and we write }A\subseteq B.$

$\displaystyle \text{Thus, if }A\subseteq B\text{ and }x\in A,\text{ then }x\in B.$

$\displaystyle \text{If }A\subseteq B,\text{ then }B\text{ is called a superset of }A,\text{ written as }B\supseteq A.$

$\displaystyle \text{Every set is a subset of itself.}$
$\displaystyle A\subseteq A,\qquad B\subseteq B.$

$\displaystyle \text{The empty set is a subset of every set.}$
$\displaystyle \phi\subseteq A,\qquad\phi\subseteq B.$

$\displaystyle \text{If }A\subseteq B\text{ and }B\subseteq A,\text{ then }A=B.$

$\displaystyle \text{Similarly, if }A=B,\text{ then }A\subseteq B\text{ and }B\subseteq A.$

$\displaystyle \text{If a set }A\text{ contains }n\text{ elements, then }A\text{ has }2^n\text{ subsets.}$

$\displaystyle \textbf{Power Set}$

$\displaystyle \text{The set of all possible subsets of a set }A\text{ is called the power set of }A$
$\displaystyle \text{and is denoted by }P(A).$

$\displaystyle \text{If }A\text{ contains }n\text{ elements, then }P(A)\text{ contains }2^n\text{ elements.}$
$\displaystyle \text{Thus, }n(P(A))=2^n.$

$\displaystyle \text{For example, if }A=\{1,2\},\text{ then }n(P(A))=2^2=4.$

$\displaystyle \text{The subsets of }A\text{ are }\phi,\{1\},\{2\}\text{ and }\{1,2\}.$

$\displaystyle \therefore P(A)=\{\phi,\{1\},\{2\},\{1,2\}\}.$

$\displaystyle \textbf{Proper Subset}$

$\displaystyle \text{Let }A\text{ and }B\text{ be sets. Then }A\text{ is called a proper subset of }B\text{ if every}$
$\displaystyle \text{element of }A\text{ is in }B,\text{ and }B\text{ has at least one element which is not in }A.$

$\displaystyle \text{A proper subset is denoted by }A\subset B.$

$\displaystyle \text{Thus, if }A\subseteq B\text{ and }A\neq B,\text{ then }A\subset B.$

$\displaystyle \text{The empty set }\phi\text{ has no proper subset.}$

$\displaystyle \text{A set containing }n\text{ elements has }2^n-1\text{ proper subsets.}$

$\displaystyle \text{For example, if }A=\{1,2,3,4\},\text{ then the number of proper subsets is}$
$\displaystyle 2^4-1=16-1=15.$

$\displaystyle \textbf{Universal Set}$

$\displaystyle \text{For a collection of sets under consideration, a set containing all the elements of these sets}$
$\displaystyle \text{is called a universal set. It is usually denoted by }U\text{ or }\xi.$

$\displaystyle \text{For example, if }A=\{1,2\},\ B=\{3,4\}\text{ and }C=\{1,5\},\text{ then we may take}$
$\displaystyle U\text{ or }\xi=\{1,2,3,4,5\}.$

$\displaystyle \textbf{Operations on Sets}$

$\displaystyle \textbf{Union of Sets}$

$\displaystyle \text{The union of sets }A\text{ and }B,\text{ denoted by }A\cup B,\text{ is the set of all elements}$
$\displaystyle \text{which belong to }A\text{ or }B\text{ or to both }A\text{ and }B.$

$\displaystyle \text{If }A=\{2,3\}\text{ and }B=\{a,b\},\text{ then }A\cup B=\{2,3,a,b\}.$

$\displaystyle A\cup B=\{x:x\in A\text{ or }x\in B\}.$

$\displaystyle x\in A\cup B\text{ means }x\in A\text{ or }x\in B.$

$\displaystyle x\notin A\cup B\text{ means }x\notin A\text{ and }x\notin B.$

$\displaystyle \textbf{Intersection of Sets}$

$\displaystyle \text{The intersection of sets }A\text{ and }B,\text{ denoted by }A\cap B,\text{ is the set of all}$
$\displaystyle \text{elements which are common to both }A\text{ and }B.$

$\displaystyle \text{If }A=\{1,2,3\}\text{ and }B=\{2,4,5\},\text{ then }A\cap B=\{2\}.$

$\displaystyle A\cap B=\{x:x\in A\text{ and }x\in B\}.$

$\displaystyle x\in A\cap B\text{ means }x\in A\text{ and }x\in B.$

$\displaystyle x\notin A\cap B\text{ means }x\notin A\text{ or }x\notin B.$

$\displaystyle \textbf{Disjoint Sets}$

$\displaystyle \text{Two sets }A\text{ and }B\text{ are called disjoint sets if they have no element in common.}$

$\displaystyle \text{Therefore, }A\cap B=\phi.$

$\displaystyle \textbf{Intersecting Sets}$

$\displaystyle \text{Two sets are said to be intersecting or overlapping if they have at least one element in common.}$

$\displaystyle \text{Therefore, }A\text{ and }B\text{ are intersecting if and only if }A\cap B\neq\phi.$

$\displaystyle \textbf{Properties of Intersection of Sets}$

$\displaystyle \text{Intersection of sets is commutative:}$
$\displaystyle A\cap B=B\cap A.$

$\displaystyle \text{Intersection of sets is associative:}$
$\displaystyle (A\cap B)\cap C=A\cap(B\cap C).$

$\displaystyle \text{If }A\subseteq B,\text{ then }A\cap B=A.$

$\displaystyle \text{Since }A\subseteq\xi,\text{ we have }A\cap\xi=A.$

$\displaystyle \text{For any sets }A\text{ and }B,\text{ we have}$
$\displaystyle A\cap B\subseteq A\text{ and }A\cap B\subseteq B.$

$\displaystyle A\cap\phi=\phi\text{ for every set }A.$

$\displaystyle \textbf{Difference of Sets}$

$\displaystyle \text{For any two sets }A\text{ and }B,\text{ the difference }A-B\text{ is the set of all those elements}$
$\displaystyle \text{of }A\text{ which are not in }B.$

$\displaystyle \text{If }A=\{1,2,3,4,5\}\text{ and }B=\{4,5,6\},\text{ then}$
$\displaystyle A-B=\{1,2,3\}\text{ and }B-A=\{6\}.$

$\displaystyle A-B=\{x:x\in A\text{ and }x\notin B\}.$

$\displaystyle x\in A-B\text{ means }x\in A\text{ and }x\notin B.$

$\displaystyle \text{If }A\subseteq B,\text{ then }A-B=\phi.$

$\displaystyle \textbf{Complement of a Set}$

$\displaystyle \text{Let }\xi\text{ be the universal set and let }A\subseteq\xi.\text{ The complement of }A,\text{ denoted}$
$\displaystyle \text{by }A',\text{ is the set of all those elements of }\xi\text{ which are not in }A.$

$\displaystyle \text{For example, let }\xi=\{1,2,3,4,5,6,7,8\}\text{ and }A=\{2,3,4\}.\text{ Then}$
$\displaystyle A'=\{1,5,6,7,8\}.$

$\displaystyle A'=\{x:x\in\xi\text{ and }x\notin A\}.$

$\displaystyle x\in A'\text{ means }x\in\xi\text{ and }x\notin A.$

$\displaystyle \text{Please note:}$
$\displaystyle \phi'=\xi\text{ and }\xi'=\phi.$
$\displaystyle A\cup A'=\xi\text{ and }A\cap A'=\phi.$

$\displaystyle \textbf{Distributive Laws for Union and Intersection of Sets}$

$\displaystyle \text{For any three sets }A,B\text{ and }C,\text{ we have:}$

$\displaystyle A\cup(B\cap C)=(A\cup B)\cap(A\cup C).$

$\displaystyle \text{For example, let }A=\{1,2\},\ B=\{2,3\}\text{ and }C=\{3,4\}.$
$\displaystyle B\cap C=\{3\}.$
$\displaystyle A\cup(B\cap C)=\{1,2,3\}.$
$\displaystyle (A\cup B)\cap(A\cup C)=\{1,2,3\}.$
$\displaystyle \therefore A\cup(B\cap C)=(A\cup B)\cap(A\cup C).$

$\displaystyle A\cap(B\cup C)=(A\cap B)\cup(A\cap C).$

$\displaystyle \text{For the same sets, }B\cup C=\{2,3,4\}.$
$\displaystyle A\cap(B\cup C)=\{2\}.$
$\displaystyle (A\cap B)\cup(A\cap C)=\{2\}.$
$\displaystyle \therefore A\cap(B\cup C)=(A\cap B)\cup(A\cap C).$