1.Relations Class 12 Examples with Solution

Relations Class 12 Examples

The concept of the term ‘relation’ Relations Class 12 Examples with Solution has been drawn from the meaning of relation in English language,according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities.

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2.HIGHLIGHTS of Types of Relations:

Relations Class 12 Examples

In this section we would like to study different types of relations.we knows that a relation in a set A is x subset of A×A.Thus,the empty set and A×A are two extreme relations.
(1.)Empty Relation:A relation R in a set A is called empty relations,if no element of A related to any element of A,i.e., R=\phi \subset A \times A
(2.)universal Relation A relation R in a set A is called universal relation,if each element of A is related to every element of A i.e.,R=A×A
Both the empty relation and the Universal relation are some times called trivial relations.
(3.)A relation R in set A is called
(i)reflexive,if (a,a) \in R for every a \in A
(ii)Symmetric,if (a_1,a_2) \in R implies that (a_2,a_1) \in R for all a_1,a_2 \in A
(iii).transitive,if (a_1,a_2) \in R and (a_2,a_3) \in R implies that (a_1,a_3) \in R for all a_1,a_2, a_3 \in A
(4.) Equivalence Relation:A relation R in a set A is said to be an equivalence relation if R is reflexive,symmetric and transitive.

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3.Relations Class 12 Examples:

Example:1.Determine whether each of the following relations are reflexive,symmetric and transitive:
Example:(1) (i).Relation R in the set A={1,2,3, …, 13,14) defined as R={(x, y):3x-y= 0}
Solution:A={1,2,3, …, 13,14) and R={(x, y):3x-y= 0}
(a)put y=x then
3x-x=2x \neq 0 \left[ \because x \neq 0 \right] \\ x \cancel{R} x \Rightarrow (x,x) \notin R \\ \therefore R is not reflexive
(b)interchanging x and y each other if
3x-y = 0 \\ \Rightarrow 3y-x \neq 0 \\ x R y \cancel{\Rightarrow} y R x \\ (x,y) \in R \cancel{\Rightarrow} (y,x) \in R \\ \therefore R is not Symmetric
(c)if 3x-y=0 and 3y-z=0
\Rightarrow 3y-z \neq 0 \\ (x,y)\in R, (y,z)\in R \\ \cancel{\Rightarrow} (x,z)\in R \\ \therefore R is not transitive
Example:1(ii).Relation R in the set N of natural numbers defined as R={(x,y):y=x+5 and x < 4}
Solution:N={1,2,3,4,……} and R={(x,y):y=x+5 and x < 4}
R={(1,6),(2,7),(3,8)}
(a)(x,x) \notin R \\ \therefore R is not reflexive
(ii)Now interchanging x and y to each other
(2,7)\in R \cancel{\Rightarrow} (7,2) \in R \\ \left( x,y \right) \in R \cancel{\Rightarrow} (y,x) \in R \\ \therefore R is not Symmetric
(c)R={(1,6),(2,7),(3,8)}
(1,6),(2,7) \in R but 6 \neq 2
(2,7),(3,8) \in R but 7 \neq 3
\left( x,y \right) \in R \cancel{\Rightarrow} (y,x) \in R \text{ then } (x,z) \notin R \\ \therefore R is not transitive
Example:1(iii).Relation R in the set A={1,2,3,4,5,6} as R={(x, y):y is divisible by x}
Solution:A={1,2,3,4,5,6}
(a)R={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
xRx so R is reflexive
(b) x R y \cancel{\Rightarrow} y R x \\ (1,2)\in R,\quad \text{but } (2,1)\notin R \\ \left( x,y \right) \in R \cancel{\Rightarrow} (y,x) \in R \\ \therefore R is not Symmetric
(c) (1,3)\in R \text{ and } (3,6) \in R\Rightarrow (1,6)\in R \\ \left( x,y \right) \in R , (y,z) \in R \Rightarrow (x,z) \in R \\ \therefore R is transitive
Example:1(iv).Relation R in the set Z of all integers defined as R={(x,y):x-y is an integer}
Solution:Z={…….. -4,-3,-2,-1,0,1,2,3,4…..} and R={(x,y):x-y is an integer}
(a)put x=y in x-y=integer
\Rightarrow x-x=0 which is integer
\therefore R is reflexive
(b)x-y=integer then y-x=integer
x R y and y R z
or \left( x,y \right) \in R \Rightarrow (y,x) \in R \\ \therefore R is Symmetric
(c)x-y=integer,y-z=integer then x-z=integer
x R y and y R z \Rightarrow x R z
\therefore R is transitive Thus R is reflexive,symmetric and transitive
Example:1(v).Relation R in the set A of human beings in a town at a particular time given by
Example:1(v)(a).R={(x,y):x and y work at the same place}
Solution:(i). (x,x) \in R \\ \therefore R is reflexive
(ii). (x,y) \in R \Rightarrow (y,x) \in R\\ \Rightarrow x R y \Rightarrow y R x
We can say x and y work at the same place then y and x work at the same place.
\therefore R is Symmetric
(iii). x R y and y R z \Rightarrow x R z
or (x,y) \in R, (y,z) \in R \Rightarrow (x,z)\in R \\ \therefore R is transitive
Example:1(v)(b). R={(x,y):x and y live in the same locality}
Solution:(i) x R x  or (x,x) \in R
R is reflexive
(ii) x R y \Rightarrow y R x
or (x,y) \in R \Rightarrow (y,x) \in R \\ \therefore R is Symmetric
(iii). (x,y) \in R \text{ and } (y,z) \in R \Rightarrow (x,z) \in R \\ \therefore R is transitive
Example:1(v)(c).R={(x,y):x is exactly 7 cm taller than y}
Solution:(i) x \cancel{R} x \text{ or } (x,x) \notin R
because x is not exactly 7cm taller than x
\therefore R is not reflexive
(ii)x R y \nRightarrow y R x
or (x,y) \in R \nRightarrow (y,x) \in R
x is exactly 7cm taller than y then y is not exactly 7cm taller then y
\therefore R is not Symmetric
(iii). x R y and y R z \nRightarrow x R z
or (x,y) \in R , (y,z) \in R \nRightarrow (x,z) \in R
x is exactly 7cm taller than y and y is exactly 7cm taller than z then x is not exactly 7cm taller then z
\therefore R is not transitive
Example:1(v)(d).R={(x,y):x is wife of y}
Solution: x \cancel{R} x  \quad \text{ or } (x,x) \notin R
because any one can not wife of self.
\therefore R is not reflexive
(ii)xRy \nRightarrow yRx or
(x,y) \in R \nRightarrow (y,x) \in R
because x is wife of y then y is not wife of x
\therefore R is not Symmetric
(iii). x R y, y R z \Rightarrow x R z
or (x,y) \in R \text{ and } (y,z) \in R \Rightarrow (x,z) \in R
\therefore R is transitive
Example:1(v)(e).R={x:x is father of y}
Solution:(i) x \cancel{R} x \quad \text{or } (x, x) \notin R \\ \therefore R is not reflexive
(ii) x R y \cancel{\Rightarrow} y R x
or (x, y) \in R \cancel{\Rightarrow} (y, x) \in R \\ \therefore R is not Symmetric
(iii)x R y and y R z \nRightarrow x R z
or (x, y) \in R, (y, z) \in R \nRightarrow (x, z) \in R \\ \therefore R is not transitive
Example:2.Show that the relation R in the set R of real numbers,defined as R=\{(a,b):a\le b^2\} is neither reflexive nor symmetric nor transitive.
Solution:(i)\forall a \in R \\ a \cancel{\leq} a^2 \\ \Rightarrow (a,a) \notin R \\ \therefore R is not reflexive
(ii) aRb \Rightarrow a \leq b^2 \\ \Rightarrow b \nleq a^2 \\ \Rightarrow b \cancel{R} a \\ \therefore(a, b) \in R \Rightarrow (b,a) \notin R \\ \therefore R is not Symmetric
For example: \left(\frac{1}{2}, 2\right) \\ \frac{1}{2} \leq 2^2 \Rightarrow \left(\frac{1}{2}, 2\right) \in R
but 2 \nleq \left(\frac{1}{2}\right)^2 \Rightarrow \left(2, \frac{1}{2}\right) \notin R
\therefore R is not Symmetric
(iii)x R y \text{ and } y R z \Rightarrow x \leq y^2
and y \leq z^2 \\ \Rightarrow x \leq z^4 \\ \Rightarrow x \nleq z^2 \\ \therefore(x, y) \in R
and (y, z) \in R \Rightarrow(x, z) \notin R \\ \therefore R is not transitive
example:4,-3,1
4 \leq(-3)^2 \text { and }(-3) \leq(1)^2
but 4 \neq 1^2
Example:3.Check whether the relation R defined in the set {1,2,3,4,5,6} as R=\{(a, b): b=a+1\} is reflexive,symmetric or transitive Solution:(i)a \neq a+1 \text{ or } a \cancel{R} a \\ \therefore R is not reflexive.
(ii)b=a+1 \cancel{\Rightarrow} a=b+1 \\ a R b \cancel{\Rightarrow} b R a
or (a, b) \in R \cancel{\Rightarrow} (b,a) \in R \\ \therefore R is not symmetric
(iii)b=a+1 and c=b+1 then c \neq a+1
(a, b) \in R (b, c) \in R \cancel{\Rightarrow} (a,c) \in R
or a R b, b R C \cancel{\Rightarrow} a R C \\ \therefore R is not transitive R is not reflexive,symmetric or transitive

Relations Class 12 Examples

Example:4.Show that the relation R in R defined as R=\{(a, b)=a \leq b\} , is reflexive and transitive but not symmetric.
Solution:R=\{(a, b)=a \leq b\}
a R a \text{ or } (a, a) \in R because
a \leq a \Rightarrow(a, a) \in R \\ \therefore R is reflexive
(ii)a \leq b \cancel{\Rightarrow} b \leq a \\ a R b \cancel{\Rightarrow} b R a \\ (a, b) \in R \cancel{\Rightarrow} (b, a) \in R \\ \therefore R is not Symmetric
for example: (2,3) \in R \quad 2 < 3 but
3 \neq 2 \quad (3,2) \notin R
(iii)a \leq b, b \leq c \Rightarrow a \leq c
a R b and b R c \Rightarrow a R c
or (a, b) \in R,(b, c) \in R \Rightarrow(a, c) \in R \\ \therefore R is not transitive
Example:5.Check whether the relation R in R defined by R=\left\{(a, b): a \leq b^3\right\} is reflexive,symmetric or transitive.
Solution:(i)R=\left\{(a, b): a \leq b^3\right\}
(i) \forall a \in R \\ a \nleq a^3 \\ \therefore a R a \text { or } \therefore (a, a) \notin R \\ \therefore R is not reflexive
Example: \frac{1}{2} \leq\left(\frac{1}{2}\right)^3 \Rightarrow \frac{1}{2} \leq \frac{1}{8} \\ \therefore\left(\frac{1}{2}, \frac{1}{2}\right) \notin R
(ii) a R b \Rightarrow a \leq b^3 \\ \Rightarrow b \nleq a^3 \\ \Rightarrow b \cancel{R} a \\ \therefore (a, b) \in R \Rightarrow(b, a) \notin R \\ \therefore R is not Symmetric
Example: \frac{1}{2} \leq 1^3 \cancel{\Rightarrow} 1 \leq\left(\frac{1}{2}\right)^3 \\ \left(\frac{1}{2}, 1\right) \in R \cancel{\Rightarrow} \left(1, \frac{1}{2}\right) \in R
(iii)a R b and b R C
\Rightarrow a \leq b^3 \text{ and } b \leq c^3 \\ \Rightarrow a \leq c^9 \\ \Rightarrow a \nleq c^3 \\ \therefore (a, b) \in R,(b, c) \in R \Rightarrow(a, c) \notin R \\ \therefore R is not transitive
Example: 3, \frac{3}{2}, \frac{5}{4} \\ 3 \leq\left(\frac{3}{2}\right)^3 \text { and } \frac{3}{2} \leq\left(\frac{5}{4}\right)^3 \\ \text{ But } 3 \nleq \left(\frac{5}{4}\right)^3 \\ \left(3, \frac{3}{2}\right) \in R,\left(\frac{3}{2}, \frac{5}{4}\right) \in R \cancel{\Rightarrow} \left(3, \frac{5}{4}\right) \in R
or  3 R \frac{3}{2} \text{ and } \frac{3}{2} R \frac{5}{4} \cancel{\Rightarrow} 3 R \frac{5}{4} \in R \\ \therefore R is the reflexive,symmetric and transitive
Example:6.Show that the relation R in the set {1,2,3} given by R={(1,2),(2,1)} is symmetric but neither reflexive nor transitive. Solution:R={(1,2),(2,1)}
(1,1),(2,3),(3,3) \notin R \\ \therefore R is not reflexive
(ii)(1,2) \in R \Rightarrow (2,1) \in R
or 1 R 2 \Rightarrow 2 R 1
R is symmetric
(iii)R is not transitive because only two elements 1,2 lie in R.
Example:7.Show that the relation in the set A of all the books in a library of a college,given by R={(x,y):x and y have same number of pages} is an equivalence relation.
Solution:Let A={x:x books in a library of a college}
R={(x,y):x and y have same number of pages}
(i)xRx
Because in any book x and x has same pages
\therefore R is reflexive
(ii)x, y \in A then (x, y) \in R \\ (x, y) \in R \Rightarrow (y, x) \in R \\ \therefore R is symmetric
(iii)x, y, z \in A and (x, y) \in R,(y,z) \in R \\ \Rightarrow (x, z) \in R \\ (x, y) \in R,(y, z) \in R \Rightarrow(x, z) \in R
R is reflexive,symmetric and transitive
Example:8.Show that the relation R in the set A={1,2,3,4,5} given by R={(a, b):|a-b| is even}, is an equivalence relation.Show that all the elements of {1,3,5} are related to each other and all the elements of {2,4} are related to each other.But no element of {1,3,5} is related to any element {2,4}.
Solution: Given R=\{(a, b):|a-b| \text{is even} \}
Where a, b \in A \text{ and } A=\{1,2,3,4,5\}
(i) |a-a|=0, which is even
(a,a) \in R or a R a
\therefore R is reflexive
(ii) (a, b) \in R then
(a, b) \in R \Rightarrow |a-b| is even
\Rightarrow |-(b-a)| is even
\Rightarrow |b-a| is even
\Rightarrow (b, a) \in R
(a, b) \in R \Rightarrow (b, a) \in R
\therefore R is symmetric
(iii) (a,b) \in R \text{ and } (b,c) \in R then
\Rightarrow |a-b| is even and |b-c| iseven
\Rightarrow a-b+b-c is also even
\Rightarrow a-c is also even
\Rightarrow |a-c| is also even
\Rightarrow (a, c) \in R
(a, b) \in R,(b, c) \in R \Rightarrow (a, c) \in R
or a R b and b R C \Rightarrow a R C
R is reftexive, symmetric and transitive.
|1-3|=|-2|=2, is even
|3-5|=|-2|=2, is even
|1-5|=|-4|=4, is even
and |2-4|=|-2|=2 is even
|4-2|=|2|=2 is even
|1-2|=1 is not even
because it is not prove given relation R=\{(a, b):|a-b| \text{is even} \}
With the above Examples,one can understand the Relations Class 12 Examples with Solution.

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4.Relations Class 12 Examples practice problems for Students:

Relations Class 12 Examples

(1.)X is the set of all points in the cartesian plane and xRy if |x|=|y|
(2.)X is the set of all straight lines in the Euclidean plane and xRy if x \bot y
Answer: (1.) Reflexive,symmetric,transitive (2.) Not reflexive,Symmetric,not transitive
By solving the above questions,you can understand the Relations Class 12 Examples with Solution well because the concept is well understood when you solve it practically.

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5.Frequently Asked Questions Related to Relations Class 12 Examples with Solution:

\begin{array}{|c|} \hline \text{**छात्र-छात्राओं से आज का सवाल**} \\ \text{*"तीन अंक बताओं जिनके घनों का योग} \\ \text{करने पर तीन अर्को की ऐसी संख्या प्राप्त } \\ \text{ हो जिसमें वही अंक हो जिनके घनों का योग }\\ \text{किया गया था।"*} \\ \text{दिनांक 25.06.2026 के प्रश्न का उत्तर:} \\ \frac{49}{6} \\ \frac{x}{7}=x-7 \Rightarrow x-\frac{x}{7}=7 \\ \Rightarrow \frac{6x}{7}=7 \Rightarrow x=\frac{49}{6} \\ \text{**Today's Questing to Students**} \\ \text{*"State three digits which,when the } \\ \text{cubes are summed, yield a three-dihit }  \\  \text{number containing the Same digits } \\ \text{whose cubes were added"*} \\ \text{Answer to Question Dated 25.06.2026:} \\ \frac{49}{6} \\ \frac{x}{7}=x-7 \Rightarrow x-\frac{x}{7}=7 \\ \Rightarrow \frac{6x}{7}=7 \Rightarrow x=\frac{49}{6} \\ \hline \end{array}
This article has been prepared by **Satyam Coaching Centre** on the **Satyam Mathematics** blog.”*