\displaystyle \textbf{1.  Definition}
\displaystyle \text{Let }A\text{ and }B\text{ be two non-empty sets. }  \text{A relation }R\text{ from }A\text{ to is defined as any }B
\displaystyle \text{subset of the Cartesian product }  A\times B.
\displaystyle \text{In other words, every relation from }A\text{ to }B  \text{ consists of ordered pairs }(a,b) \text{ where }
\displaystyle a\in A \text{ and }b\in B.\ \text{Thus,}
\displaystyle R\text{ is a relation from }A\text{ to }B \Leftrightarrow R\subseteq A\times B  \Leftrightarrow R\subseteq \{(a,b):a\in A,\ b\in B\}

\displaystyle \textbf{Illustrations :}

\displaystyle \text{(a) Let }A=\{1,2,4\}\text{ and }B=\{4,6\}.
\displaystyle \text{Consider }R=\{(1,4),(1,6),(2,4),(2,6),(4,4),(4,6)\}.
\displaystyle \text{Since every ordered pair of }R\text{ belongs to }A\times B,\ \text{we have }R\subseteq A\times B.
\displaystyle \text{Therefore, }R\text{ is a relation from }A\text{ to }B.

\displaystyle \text{(b) Let }A=\{1,2,3\}\text{ and }B=\{2,3,5,7\}.
\displaystyle \text{Consider }R=\{(2,3),(3,5),(5,7)\}.
\displaystyle \text{Here, the ordered pair }(5,7)\text{ does not belong to }A\times B\text{ because }5\notin A.
\displaystyle \text{Hence, }R\nsubseteq A\times B\text{ and therefore }R\text{ is not a relation from }A\text{ to }B.

\displaystyle \text{(c) Let }A=\{-1,1,2\}\text{ and }B=\{1,4,9,10\}.
\displaystyle \text{Define a relation }R\text{ such that }aRb\text{ whenever }a^2=b.\ \text{Then, the relation is}
\displaystyle R=\{(-1,1),(1,1),(2,4)\}.

\displaystyle \textbf{Note :}

  • \displaystyle \text{A relation from }A\text{ to }B\text{ is also referred to as a relation from }A\text{ into }B.
  • \displaystyle  (a,b)\in R\text{ can also be written as }aRb,\ \text{which is read as ``}a\text{ is related to }b\text{''.}
  • \displaystyle \text{Suppose }A\text{ and }B\text{ are two non-empty finite sets containing } p \text{ and }q\text{ elements} 
    \displaystyle \text{respectively. Then }n(A\times B)=n(A)n(B)=pq. \text{Therefore, the Cartesian product }
    \displaystyle A\times B \text{ has }2^{pq}\text{ subsets. } \text{Since every subset of }A\times B\text{ defines a relation from }
    \displaystyle A\text{ to }B,\text{the total number of possible relations from }A\text{ to }B\text{ is }2^{pq}.

\displaystyle \textbf{2. Domain and Range of a Relation}

\displaystyle \textbf{(a) Domain of a Relation :}\ \text{Consider a relation }R\text{ from }A\text{ to }B.
\displaystyle \text{The domain of }R\text{ consists of all elements }a\in A\text{ for which there exists at least one }
\displaystyle \text{element }b\in B\text{ such that }(a,b)\in R.  \text{ Hence,}
\displaystyle \mathrm{Dom.}(R)=\{a\in A:(a,b)\in R\text{ for some }b\in B\}
\displaystyle \text{In simple terms, the domain is the collection of all first components appearing in }
\displaystyle \text{the ordered pairs of }R.

\displaystyle \textbf{(b) Range of a Relation :}\ \text{Let }R\text{ be a relation from }A\text{ to }B.
\displaystyle \text{The range of }R\text{ is the set of all elements }b\in B\text{ for which there exists at least one }
\displaystyle \text{element }a\in A\text{ such that }(a,b)\in R. \text{ Thus,}
\displaystyle \mathrm{Range}(R)=\{b\in B:(a,b)\in R\text{ for some }a\in A\}
\displaystyle \text{That is, the range contains all second components appearing } \text{in the ordered pairs of }R.

\displaystyle \textbf{(c) Co-domain of a Relation :}\ \text{If }R\text{ is a relation from }A\text{ to }B, \text{then the set }B
\displaystyle \text{ is called the co-domain of }R \text{Therefore, the co-domain represents the complete target set}
\displaystyle \text{associated with the relation.}

\displaystyle \textbf{Illustrations :}

\displaystyle \text{(i) Let }A=\{1,2,3,7\}\text{ and }B=\{3,6\}.
\displaystyle \text{Define a relation }R\text{ such that }aRb\text{ whenever }a<b. \text{ Then,}
\displaystyle R=\{(1,3),(1,6),(2,3),(2,6),(3,6)\}.
\displaystyle \text{For this relation, }\mathrm{Dom.}(R)=\{1,2,3\}\text{ and }\mathrm{Range}(R)=\{3,6\}.
\displaystyle \text{The co-domain of }R\text{ is }B=\{3,6\}.

\displaystyle \text{(ii) Let }A=\{1,2,3\}\text{ and }B=\{2,4,6,8\}.
\displaystyle \text{Consider the relations }R_1=\{(1,2),(2,4),(3,6)\}\text{ and } R_2=\{(2,4),(2,6),(3,8),(1,6)\}.
\displaystyle \text{Both }R_1\text{ and }R_2\text{ are relations from }A\text{ to }B  \text{ because every ordered pair belongs to } \\ A\times B.
\displaystyle \text{Here, }\mathrm{Dom.}(R_1)=\{1,2,3\},\ \mathrm{Range}(R_1)=\{2,4,6\}
\displaystyle \text{and }\mathrm{Dom.}(R_2)=\{1,2,3\},\ \mathrm{Range}(R_2)=\{4,6,8\}.

\displaystyle \textbf{3. Types of Relations}

\displaystyle \textbf{(a) Empty Relation :}\ \text{A relation }R\text{ from }A\text{ to }B\text{ is called an empty relation if it }
\displaystyle \text{contains no ordered pair, i.e., }R=\phi.
\displaystyle \text{For example, let }A=\{2,4,6\}\text{ and }B=\{7,11\}.
\displaystyle \text{Define }R=\{(a,b):a\in A,\ b\in B\text{ and }a-b\text{ is even}\}.
\displaystyle \text{Since no such ordered pair exists, }R=\phi.\ \text{Hence, }R\text{ is an empty relation.}

\displaystyle \textbf{(b) Universal Relation :}\ \text{A relation }R\text{ from }A\text{ to }B \text{ is called a universal relation when }
\displaystyle R=A\times B.
\displaystyle \text{For example, let }A=\{1,2\}\text{ and }B=\{1,3\}.
\displaystyle \text{Take }R=\{(1,1),(1,3),(2,1),(2,3)\}.
\displaystyle \text{Since every possible ordered pair from }A\times B\text{ appears in }R,  \text{ the relation }R\text{ is universal.}

\displaystyle \textbf{Note :}

  • \displaystyle \text{The empty relation }\phi\text{ and the universal relation }A\times A \text{ on a set }A \text{ are respectively the}
    \displaystyle \text{smallest and largest relations defined on }A.
  • \displaystyle  \text{These are commonly known as trivial relations, whereas all other relations are called }
    \displaystyle \text{non-trivial relations.}
  • \displaystyle  \text{The relations }R=\phi\text{ and }R=A\times A\text{ are also referred } \text{to as extreme relations.}

\displaystyle \textbf{(c) Identity Relation :}\ \text{A relation }R\text{ defined on a set }A \text{ is called an identity relation}
\displaystyle \text{if every element is related only to itself. } \text{Mathematically,}
\displaystyle R=\{(a,b):a\in A,\ b\in A\text{ and }a=b\}
\displaystyle \text{or equivalently,}
\displaystyle R=\{(a,a):a\in A\}
\displaystyle \text{The identity relation on }A\text{ is denoted by }I_A.
\displaystyle \text{For example, if }A=\{1,2,3,4\},\text{ then }
\displaystyle I_A=\{(1,1),(2,2),(3,3),(4,4)\}.
\displaystyle \text{However, the relation }R=\{(1,1),(2,2),(1,3),(4,4)\}
\displaystyle \text{is not an identity relation because }1\text{ and }3\text{ are not related to themselves.}

\displaystyle \textbf{Note :}

  • \displaystyle  \text{In an identity relation, every element of the set must be related only to itself and to no }
    \displaystyle \text{other element.}

\displaystyle \textbf{(d) Reflexive Relation :}\ \text{A relation }R\text{ on a set }A \text{is said to be reflexive if every }
\displaystyle \text{element of }A\text{ is related to itself.}
\displaystyle \text{That is, }(a,a)\in R\text{ for all }a\in A.
\displaystyle \text{For example, let }A=\{1,2,3\}\text{ and consider the relations}
\displaystyle R_1=\{(1,1),(2,2),(3,3)\},
\displaystyle R_2=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3)\}\text{ and}
\displaystyle R_3=\{(2,2),(2,3),(3,2),(1,1)\}.
\displaystyle \text{Here, }R_1\text{ and }R_2\text{ are reflexive relations on }A \text{ because every element of }A
\displaystyle \text{is related to itself.} \text{However, }R_3\text{ is not reflexive since }(3,3)\notin R_3.

\displaystyle \textbf{Note :}

  • \displaystyle  \text{The universal relation on a non-empty set is always reflexive.}
  • \displaystyle  \text{Every identity relation is reflexive, but every reflexive relation need not be an}
    \displaystyle \text{identity relation. In the above examples, }R_1\text{ is both identity } \text{and reflexive, whereas }
    \displaystyle R_2 \text{ is reflexive but not identity.}

\displaystyle \textbf{(e) Symmetric Relation :}\ \text{A relation }R\text{ defined on a set }A \text{is said to be symmetric if}
\displaystyle \text{whenever }(a,b)\in R,\text{ then }(b,a)\in R.
\displaystyle \text{In other words, if }aRb,\text{ then }bRa.
\displaystyle \text{For example, let }A=\{1,2,3\}\text{ and consider the relations}
\displaystyle R_1=\{(1,2),(2,1)\},\ R_2=\{(1,2),(2,1),(1,3),(3,1)\},
\displaystyle R_3=\{(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}
\displaystyle \text{and }R_4=\{(1,3),(3,1),(2,3)\}.
\displaystyle \text{Here, }R_1,\ R_2\text{ and }R_3\text{ are symmetric relations on }A.
\displaystyle \text{However, }R_4\text{ is not symmetric because }(2,3)\in R_4  \text{ but }(3,2)\notin R_4.

\displaystyle \textbf{(f) Transitive Relation :}\ \text{A relation }R\text{ on a set }A \text{ is called transitive if whenever }
\displaystyle (a,b)\in R\text{ and }(b,c)\in R, \text{then }(a,c)\in R.
\displaystyle \text{Equivalently, if }aRb\text{ and }bRc,\text{ then }aRc.
\displaystyle \text{For example, let }A=\{1,2,3\}\text{ and consider the relations}
\displaystyle R_1=\{(1,2),(2,3),(1,3),(3,2)\}
\displaystyle \text{and }R_2=\{(1,3),(3,2),(1,2)\}.
\displaystyle \text{Here, }R_2\text{ is transitive.}
\displaystyle \text{However, }R_1\text{ is not transitive because }(2,3)\in R_1 \text{and }(3,2)\in R_1, \text{ but }(2,2)\notin R_1.

\displaystyle \textbf{(g) Equivalence Relation :}\ \text{Let }A\text{ be a non-empty set.}
\displaystyle \text{A relation }R\text{ on }A\text{ is called an equivalence relation if it is reflexive, symmetric and }
\displaystyle \text{transitive. That is,}
\displaystyle \text{(i) }R\text{ is reflexive, i.e., }(a,a)\in R\ \forall\ a\in A.
\displaystyle \text{(ii) }R\text{ is symmetric, i.e., if }(a,b)\in R,\text{ then }(b,a)\in R.
\displaystyle \text{(iii) }R\text{ is transitive, i.e., if }(a,b)\in R\text{ and }(b,c)\in R, \text{ then }(a,c)\in R.
\displaystyle \text{For example, let }A=\{1,2,3\}\text{ and}
\displaystyle R=\{(1,1),(1,2),(2,1),(2,2),(3,3),(1,3),(3,1),(2,3),(3,2)\}.
\displaystyle \text{Since }R\text{ is reflexive, symmetric and transitive, } \text{it is an equivalence relation on }A.

\displaystyle \textbf{Equivalence Classes :}\ \text{Let }R\text{ be an equivalence relation on a set }A \text{ and let }
\displaystyle a\in A. \text{ Then, the set of all elements of }A\text{ related to }a\text{ is called the equivalence class }
\displaystyle \text{determined by }a\text{ and is denoted by }[a].
\displaystyle \text{Thus,} \  [a]=\{b\in A:(a,b)\in R\}

\displaystyle \textbf{Note :}

  • \displaystyle  \text{Two equivalence classes are either identical or disjoint.}
  • \displaystyle  \text{An equivalence relation on a set partitions the set into mutually disjoint }
    \displaystyle \text{equivalence classes.}
  • \displaystyle  \text{A key feature of an equivalence relation is that it divides a set into pairwise }
    \displaystyle \text{disjoint subsets called equivalence classes.}
  • \displaystyle  \text{The collection of all equivalence classes forms a partition of the set.}
  • \displaystyle  \text{The union of all equivalence classes gives back the entire set.}

\displaystyle \text{For example, let }R\text{ be the relation on integers defined by } \\ R=\{(a,b):2\text{ divides }a-b\}.
\displaystyle \text{Then, the equivalence class }[0]\text{ is } [0]=\{0,\pm2,\pm4,\pm6,\ldots\}

\displaystyle \textbf{4. Tabular Representation of a Relation}

\displaystyle \text{In tabular form, the elements of set }A\text{ are written as row headings } \text{and the elements of set }
\displaystyle B\text{ are written as column headings.}
\displaystyle \text{If }(a,b)\in R,\text{ then we write }1\text{ in the corresponding position;}  \text{otherwise, we write }0.
\displaystyle \text{For example, let }A=\{1,2,3\},\ B=\{2,5\} \text{ and }R=\{(1,2),(2,5),(3,2)\}.
\displaystyle \text{Then, the tabular representation of }R\text{ is}
\displaystyle \begin{array}{|c|c|c|} \hline R & 2 & 5 \\ \hline 1 & 1 & 0 \\ \hline 2 & 0 & 1 \\ \hline 3 & 1 & 0 \\ \hline \end{array}

\displaystyle \textbf{5. Inverse Relation}

\displaystyle \text{Let }R\subseteq A\times B\text{ be a relation from }A\text{ to }B.
\displaystyle \text{The inverse relation of }R,\text{ denoted by }R^{-1}, \text{is a relation from }B\text{ to }A \text{ defined by }
\displaystyle R^{-1}=\{(b,a):(a,b)\in R\}.
\displaystyle \text{Thus, }(a,b)\in R\Leftrightarrow (b,a)\in R^{-1}.
\displaystyle \text{Clearly, }\mathrm{Dom.}(R^{-1})=\mathrm{Range}(R)  \text{ and }\mathrm{Range}(R^{-1})=\mathrm{Dom.}(R).
\displaystyle \text{Also, }(R^{-1})^{-1}=R.
\displaystyle \text{For example, let }A=\{1,2,4\},\ B=\{3,0\} \text{ and let }R=\{(1,3),(4,0),(2,3)\}
\displaystyle \text{ be a relation from }A\text{ to }B.
\displaystyle \text{Then, }  R^{-1}=\{(3,1),(0,4),(3,2)\}.

\displaystyle \textbf{Know the Facts}

\displaystyle \textbf{1. }\text{(i) A relation }R\text{ from }A\text{ to }B\text{ is called an empty} \text{ or void relation if }R=\phi.
\displaystyle \text{(ii) A relation }R\text{ on a set }A\text{ is called an empty or void }  \text{relation if }R=\phi.

\displaystyle \textbf{2. }\text{(i) A relation }R\text{ from }A\text{ to }B\text{ is called a universal }  \text{relation if }R=A\times B.
\displaystyle \text{(ii) A relation }R\text{ on a set }A\text{ is called a universal } \text{relation if }R=A\times A.

\displaystyle \textbf{3. }\text{A relation }R\text{ on a set }A\text{ is reflexive if }aRa  \text{f or every }a\in A.

\displaystyle \textbf{4. }\text{A relation }R\text{ on a set }A\text{ is symmetric if}  \text{whenever }aRb,\text{ then }bRa\text{ for all } \\ a,b\in A.

\displaystyle \textbf{5. }\text{A relation }R\text{ on a set }A\text{ is transitive if whenever}  aRb\text{ and }bRc,\text{ then }aRc \\ \text{ for all }a,b,c\in A.

\displaystyle \textbf{6. }\text{A relation }R\text{ on }A\text{ is an identity relation if } R=\{(a,a):a\in A\},
\displaystyle \text{ i.e., }R\text{ contains only ordered pairs of the form }  (a,a)\text{ and no other elements.}

\displaystyle \textbf{7. }\text{A relation }R\text{ on a non-empty set }A\text{ is called an equivalence relation if }
\displaystyle \text{the following conditions are satisfied :}

\displaystyle \text{(i) }R\text{ is reflexive, i.e., }(a,a)\in R\text{ for every }a\in A.
\displaystyle \text{(ii) }R\text{ is symmetric, i.e., if }(a,b)\in R,\text{ then }(b,a)\in R.
\displaystyle \text{(iii) }R\text{ is transitive, i.e., if }(a,b)\in R\text{ and }(b,c)\in R,
\displaystyle \text{then }(a,c)\in R.

\displaystyle \textbf{Types of Intervals}

\displaystyle \textbf{(i) Open Interval :}\ \text{If }a\text{ and }b\text{ are real numbers such that }a<b,
\displaystyle \text{then the set of all real numbers lying strictly between }a\text{ and }b \text{ is called an open interval.}
\displaystyle \text{It is denoted by }]a,b[\text{ or }(a,b), \text{ i.e., }\{x\in R:a<x<b\}.

\displaystyle \textbf{(ii) Closed Interval :}\ \text{If }a<b,\text{ then the set of all real numbers } \text{lying between }
\displaystyle a\text{ and }b,\text{ including both endpoints,}  \text{ is called a closed interval.}
\displaystyle \text{It is denoted by }[a,b], \text{ i.e., }\{x\in R:a\leq x\leq b\}.

\displaystyle \textbf{(iii) Open Closed Interval :}\ \text{If }a<b,\text{ then the set of all real }\text{numbers between }
\displaystyle a\text{ and }b\text{ excluding }a\text{ but including }b  \text{ is called an open closed interval.}
\displaystyle \text{It is denoted by }]a,b]\text{ or }(a,b],  \text{i.e., }\{x\in R:a<x\leq b\}.

\displaystyle \textbf{(iv) Closed Open Interval :}\ \text{If }a<b,\text{ then the set of all real numbers between}
\displaystyle  a\text{ and }b\text{ including }a\text{ but excluding }b  \text{ is called a closed open interval.}
\displaystyle \text{It is denoted by }[a,b[\text{ or }[a,b),  \text{ i.e., }\{x\in R:a\leq x<b\}.

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