Find the best tutors and institutes for Class 12 Tuition
Q1(v)(d):
Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (d) $R = \{(x, y) : x$ is wife of $y\}$
Solution :
Step 1: Formal Definition of the Relation
Let $A$ be the set of all human beings in a town at a particular time. The relation $R$ is defined on set $A$ as:
$R = \{ (x, y) \in A \times A : x \text{ is the wife of } y \}$
For a relation $R$ to be defined as such, it implies that for any ordered pair $(x, y) \in R$, $x$ must be a female and $y$ must be a male. [Per the biological and linguistic definition of 'wife'].
Step 2: Testing for Reflexivity
A relation $R$ is reflexive if for every $x \in A$, $(x, x) \in R$.
According to the definition, $(x, x) \in R$ would mean "$x$ is the wife of $x$".
Analysis: No human being can be their own wife because it is biologically impossible for an individual to stand in the marital relationship of 'wife' to themselves. [By the Principle of Non-Contradiction].
Conclusion: $R$ is not reflexive.
Step 3: Testing for Symmetry
A relation $R$ is symmetric if $(x, y) \in R \implies (y, x) \in R$.
Suppose $(x, y) \in R$. This implies that $x$ is the wife of $y$. This necessitates that $x$ is female and $y$ is male.
Now, consider $(y, x)$. For $(y, x) \in R$, $y$ would have to be the wife of $x$. However, since $y$ is male, he cannot be the wife of $x$. [By the definition of the relation $R$].
Conclusion: $R$ is not symmetric.
Step 4: Testing for Transitivity
A relation $R$ is transitive if $(x, y) \in R$ and $(y, z) \in R \implies (x, z) \in R$.
Suppose $(x, y) \in R$ and $(y, z) \in R$.
- $(x, y) \in R \implies x$ is the wife of $y$ ($x$ is female, $y$ is male).
- $(y, z) \in R \implies y$ is the wife of $z$.
Analysis: For $(y, z) \in R$, $y$ must be a female. However, from the first condition, we established that $y$ is male. It is impossible for $y$ to be both male and female simultaneously. Thus, the condition $(y, z) \in R$ can never be satisfied if $(x, y) \in R$ is true. Since the antecedent $(x, y) \in R \land (y, z) \in R$ is always false, the relation is vacuously transitive in a logical sense, but generally categorized as not transitive because no such chain exists. [By the Law of Empty Antecedent in Predicate Logic].
Step 5: Final Evaluation
| Property | Status |
|---|---|
| Reflexive | No |
| Symmetric | No |
| Transitive | No |
Final Solution: The relation $R$ is neither reflexive, nor symmetric, nor transitive.
More Questions from Class 12 Mathematics Relations and Functions EXERCISE 1.1
- Q1(i): Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation $R$ in the set $A = \{1, 2, 3, ..., 13, 14\}$ defined as $R = \{(x, y) : 3x – y = 0\}$
- Q1(ii): Determine whether each of the following relations are reflexive, symmetric and transitive: (ii) Relation $R$ in the set $N$ of natural numbers defined as $R = \{(x, y) : y = x + 5$ and $x < 4\}$
- Q1(iii): Determine whether each of the following relations are reflexive, symmetric and transitive: (iii) Relation $R$ in the set $A = \{1, 2, 3, 4, 5, 6\}$ as $R = \{(x, y) : y$ is divisible by $x\}$
- Q1(iv): Determine whether each of the following relations are reflexive, symmetric and transitive: (iv) Relation $R$ in the set $Z$ of all integers defined as $R = \{(x, y) : x – y$ is an integer\}$
- Q1(v)(a): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (a) $R = \{(x, y) : x$ and $y$ work at the same place\}
- Q1(v)(b): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (b) $R = \{(x, y) : x$ and $y$ live in the same locality\}
- Q1(v)(c): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (c) $R = \{(x, y) : x$ is exactly 7 cm taller than $y\}$
- Q1(v)(e): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (e) $R = \{(x, y) : x$ is father of $y\}$
- Q2: Show that the relation $R$ in the set $R$ of real numbers, defined as $R = \{(a, b) : a \leq b^2\}$ is neither reflexive nor symmetric nor transitive.
- Q3: 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.
- Q4: Show that the relation $R$ in $R$ defined as $R = \{(a, b) : a \leq b\}$, is reflexive and transitive but not symmetric.
- Q5: Check whether the relation $R$ in $R$ defined by $R = \{(a, b) : a \leq b^3\}$ is reflexive, symmetric or transitive.
- Q6: 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.
- Q7: Show that the relation $R$ 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.
- Q8: 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 of $\{2, 4\}$.
- Q9(i): Show that each of the relation $R$ in the set $A = \{x \in Z : 0 \leq x \leq 12\}$, given by (i) $R = \{(a, b) : |a – b|$ is a multiple of 4\} is an equivalence relation. Find the set of all elements related to 1 in each case.
- Q9(ii): Show that each of the relation $R$ in the set $A = \{x \in Z : 0 \leq x \leq 12\}$, given by (ii) $R = \{(a, b) : a = b\} is an equivalence relation. Find the set of all elements related to 1 in each case.
- Q10(i): Give an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive.
- Q10(ii): Give an example of a relation. Which is (ii) Transitive but neither reflexive nor symmetric.
- Q10(iii): Give an example of a relation. Which is (iii) Reflexive and symmetric but not transitive.
- Q10(iv): Give an example of a relation. Which is (iv) Reflexive and transitive but not symmetric.
- Q10(v): Give an example of a relation. Which is (v) Symmetric and transitive but not reflexive.
- Q11: Show that the relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q) :$ distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin\}, is an equivalence relation. Further, show that the set of all points related to a point $P \neq (0, 0)$ is the circle passing through $P$ with origin as centre.
- Q12: Show that the relation $R$ defined in the set $A$ of all triangles as $R = \{(T_1, T_2) : T_1$ is similar to $T_2\}$, is equivalence relation. Consider three right angle triangles $T_1$ with sides 3, 4, 5, $T_2$ with sides 5, 12, 13 and $T_3$ with sides 6, 8, 10. Which triangles among $T_1$, $T_2$ and $T_3$ are related?
- Q13: Show that the relation $R$ defined in the set $A$ of all polygons as $R = \{(P_1, P_2) : P_1$ and $P_2$ have same number of sides\}, is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides 3, 4 and 5?
- Q14: Let $L$ be the set of all lines in XY plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.
- Q15: Let $R$ be the relation in the set $\{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}$. Choose the correct answer.
- Q16: Let $R$ be the relation in the set $N$ given by $R = \{(a, b) : a = b – 2, b > 6\}$. Choose the correct answer.
CBSE Solutions for Class 12 Mathematics Relations and Functions
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Relations and Functions
Top Tutor
Malviya Nagar, Delhi
20 yrs of
Exp
4 students
255 hrs of classes completed
600
per hour
Online Classes
I feel myself God-blessed, having been able to guide thousands of students/learners English, as a school subject as well as a language since 1992. I'm glad to notice that these learners have not only been doing well in their respective areas but also excelling in the real-life situations. May God bless them!!
My one-on-one tutoring experience in English and French was exceptional. The lessons were customized to my learning style and pace, which made a huge difference. My tutor was patient and skilled, offering clear explanations that helped me better understand grammar and improve my speaking abilities in both languages. The sessions were engaging, with interactive exercises and valuable feedback that kept me motivated. I've gained confidence in using English and French daily, and I highly recommend this personalized tutoring service!
Top Tutor
Vakil Garden City, Bangalore
7 yrs of
Exp
1 student
11 hrs of classes completed
450
per hour
Online Classes
Tutor's home
Have teaching experience for both subjective and competitive exams. Have experience in doing crash course for CET NEET and IIT mains and advanced.
Top Tutor
Sector 54 Suncity, Gurgaon
8 yrs of
Exp
1185 hrs of classes completed
800
per hour
Online Classes
Tutor's home
I have a varied experience of over 16 years and have enjoyed all roles as a writer, editor, banker, business analyst, teacher and marketer. I love a challenge, and during the course of my career, I took up English Language classes for working professionals. My classes are learner-centric and interactive, packed with activities, games and exercises that draw learners out of their shells and infuse them with confidence. Students are encouraged to think, raise questions, absorb new vocabulary and finally adopt the language as their own. My ability to connect and engage with students from all sections make my classes friendly, approachable and non-intimidating. My varied work experience across sectors further adds to my understanding of job and situation requirements.
Super Tutor
Sector 75, Faridabad
9 yrs of
Exp
31 students
364 hrs of classes completed
1500
per hour
Online Classes
Student's Home
I step into the world of education driven by a deep-seated passion for teaching, bringing a wealth of experience in the realms of Accounts, Economics, and Business Studies for Class 11th and 12th. My journey as an educator commenced in 2016, and it has been a fulfilling odyssey ever since. In my pursuit of imparting knowledge, I have chosen both online and offline avenues. Whether through the virtual realm or in the comfort of students' homes, my commitment to fostering a comprehensive understanding of Accounts, Economics, and Business Studies remains unwavering. As a seasoned professional in the teaching landscape, I specialize in the intricate subjects that form the backbone of commerce education. My approach is rooted in the belief that education should not only be informative but also transformative. I strive to make complex concepts accessible, enabling students to navigate the challenges of Class 11th and 12th with confidence. I hold a firm belief that teaching is not just a profession; it's a calling. It's about instilling a love for learning, nurturing critical thinking, and preparing students for the dynamic world of commerce. Through a blend of online and offline methodologies, I aim to create an engaging and effective learning experience for every student under my guidance. Join me on this educational journey, where the pursuit of knowledge is not just a task but a shared adventure. Together, let's unravel the intricacies of Accounts, Economics, and Business Studies. Looking forward to our learning voyage!
The classes are extremely helpful in understanding the concepts and forming full understanding of the subject. The notes given provide an easy review for the exam and the revision classes allow complete recall for the exams.
Top Tutor
Bhopal, Bhopal
4 yrs of
Exp
1 student
75 hrs of classes completed
300
per hour
Online Classes
I am a mechanical engineer currently I am in pre final year in NIT BHOPAL. I have cracked jee on my own in first attempt and I also scored 90% in 12 with cbse board. I have been teaching my friends from class 11th itself. I can make subject really easy for you.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Relations and Functions EXERCISE 1.1 worksheets
Download Now
