Find the best tutors and institutes for Class 11 Tuition
Q4(iii):
Show that the following four conditions are equivalent : (iii) $A \cup B = B$
Solution :
Theoretical Foundation & Logical Equivalence
To demonstrate that a condition $A \cup B = B$ is equivalent to $A \subseteq B$, we must prove the bidirectional implication (biconditional statement): $A \subseteq B \iff A \cup B = B$. This relies on the Axioms of Set Theory, specifically the definitions of Union and Subset relations.
Step 1: Forward Implication ($\implies$) - Proving $A \subseteq B \implies A \cup B = B$
Assume $A \subseteq B$. We aim to show that $A \cup B = B$ by demonstrating mutual inclusion ($A \cup B \subseteq B$ and $B \subseteq A \cup B$).
- Part A: Show $A \cup B \subseteq B$
Let $x \in A \cup B$. By the definition of union, $x \in A$ or $x \in B$ [Definition of Set Union].
Case 1: If $x \in A$, then since $A \subseteq B$, it follows that $x \in B$ [Definition of Subset].
Case 2: If $x \in B$, the condition is trivially satisfied.
Thus, $x \in B$ in all cases, confirming $A \cup B \subseteq B$. - Part B: Show $B \subseteq A \cup B$
For any set $B$, it is a fundamental property that any element belonging to $B$ also belongs to the union of $B$ with any set $A$ [Identity/Monotonicity of Union].
Formally, if $y \in B$, then $y \in B$ or $y \in A$, which implies $y \in A \cup B$.
Thus, $B \subseteq A \cup B$.
By the Axiom of Extensionality, since $A \cup B \subseteq B$ and $B \subseteq A \cup B$, we conclude $A \cup B = B$.
Step 2: Reverse Implication ($\impliedby$) - Proving $A \cup B = B \implies A \subseteq B$
Assume $A \cup B = B$. We aim to show $A \subseteq B$.
- Let $x$ be an arbitrary element such that $x \in A$.
- By the property of union, $x \in A \cup B$ [Definition of Union].
- Since we are given that $A \cup B = B$, it follows by substitution that $x \in B$.
- Since $x \in A$ implies $x \in B$, it follows by definition that $A \subseteq B$ [Definition of Subset].
Step 3: Visual Representation (Venn Diagram)
Summary of Logic
| Logical Direction | Process | Conclusion |
|---|---|---|
| $A \subseteq B \implies A \cup B = B$ | Subset Inclusion | Verified via Element-wise Analysis |
| $A \cup B = B \implies A \subseteq B$ | Identity Substitution | Verified via Definition of Union |
Final Solution: Since the forward and reverse implications hold true, it is rigorously established that $A \cup B = B$ is logically equivalent to $A \subseteq B$.
More Questions from Class 11 Mathematics Sets Miscellaneous Exercise on Chapter 1
- Q1: Decide, among the following sets, which sets are subsets of one and another: $A = \{ x : x \in R$ and $x$ satisfy $x^2 – 8x + 12 = 0 \}$, $B = \{ 2, 4, 6 \}$, $C = \{ 2, 4, 6, 8, . . . \}$, $D = \{ 6 \}$.
- Q2(i): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (i) If $x \in A$ and $A \in B$, then $x \in B$
- Q2(ii): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (ii) If $A \subset B$ and $B \in C$, then $A \in C$
- Q2(iii): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (iii) If $A \subset B$ and $B \subset C$, then $A \subset C$
- Q2(iv): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (iv) If $A \not\subset B$ and $B \not\subset C$, then $A \not\subset C$
- Q2(v): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (v) If $x \in A$ and $A \not\subset B$, then $x \in B$
- Q2(vi): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (vi) If $A \subset B$ and $x \notin B$, then $x \notin A$
- Q3: Let $A, B$, and $C$ be the sets such that $A \cup B = A \cup C$ and $A \cap B = A \cap C$. Show that $B = C$.
- Q4(i): Show that the following four conditions are equivalent : (i) $A \subset B$
- Q4(ii): Show that the following four conditions are equivalent : (ii) $A – B = \phi$
- Q4(iv): Show that the following four conditions are equivalent : (iv) $A \cap B = A$
- Q5: Show that if $A \subset B$, then $C – B \subset C – A$.
- Q6: Show that for any sets $A$ and $B$, $A = ( A \cap B ) \cup ( A – B )$ and $A \cup ( B – A ) = ( A \cup B )$
- Q7(i): Using properties of sets, show that (i) $A \cup ( A \cap B ) = A$
- Q7(ii): Using properties of sets, show that (ii) $A \cap ( A \cup B ) = A$.
- Q8: Show that $A \cap B = A \cap C$ need not imply $B = C$.
- Q9: Let $A$ and $B$ be sets. If $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$ for some set $X$, show that $A = B$. (Hints $A = A \cap ( A \cup X ) , B = B \cap ( B \cup X )$ and use Distributive law )
- Q10: Find sets $A, B$ and $C$ such that $A \cap B, B \cap C$ and $A \cap C$ are non-empty sets and $A \cap B \cap C = \phi$.
CBSE Solutions for Class 11 Mathematics Sets
Chapters in CBSE - Class 11 Mathematics
Top Tutors who teach Sets
Top Tutor
Tilakwadi, Belgaum
6 yrs of
Exp
10 students
1000 hrs of classes completed
400
per hour
Online Classes
I am a passion driven tutor, with a rich experience of teaching students of intermediate level, Physics, Chemistry and Mathematics, in a real classroom environment at my residence, for nearly 6 years. I have taught about 10 to 16 students every year, during this period I used to enliven the students through practical examples from my field of service, to develop their fundamental concepts/interest and enthuse them to gain a deeper insight into the subject. I used to display reputed books of foreign authors to portray the concepts, through explanatory photos and diagrams, in the absence of electronic educational aids during that period (1993 to 1998). However, with the present state of Technology, invaluable resources from prestigious Universities/Institutions and the excellent facilities available on the UrbanPro Live Platform, I am sure, the students will have a rich learning experience I am very much a down to earth personality, with communication/conversational skills, adequate to engage with learners possessing varied interests. This skill enables me to interact with/counsel young learners to strengthen their skills to realize their various ambitions in Life. I introduce myself as an Engineering graduate with a Track of Professional experience in the field of Telecommunication, Power and Geo-sciences, with a brief supporting role in a Technological University, totally spanning over four decades till date. I will be organizing a free Demo Class for prospective students, who wish to apprise themselves of my personality and teaching methods. The schedule of the same will be put up on the UrbanPro Live Platform
Mukundan Sir is very knowledgeable and a great person. He teaches the concepts very well, gives extra exercises for better understanding. Never know how 1 hour of class goes so quickly as I get immersed with his teaching. Fortunate to have you as my teacher Sir. Thank you so much.
Top Tutor
Knowledge Park I, Noida
20 yrs of
Exp
17 students
4155 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I am principal in a cbse school from last 4 years in greater noida before this I was PGT Accountancy in school of ghaziabad. I m in teaching since 1997 and author of accountancy book exam guru,super 20 sample paper and chapterwise 10 years solved papers of full marks pvt limited. I started this in 2015.
Such a wonderful way of teaching which is helpful for understanding every variety of questions. Never feel bored in my study time.
Top Tutor
Sector 61 Block E, Noida
20 yrs of
Exp
2 students
1158 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I have extensive teaching experience providing personalized 1:1 Python training to students. My teaching approach focuses on building strong Python fundamentals through interactive lessons, practical examples, and regular coding practice. In 1:1 sessions, I adapt my teaching methods to suit each student's learning pace and individual needs, ensuring concepts are clearly understood. I encourage logical thinking, problem-solving, and hands-on coding, helping students gain confidence, strengthen their Python programming skills, and achieve academic and professional success.
Top Tutor
Beta I Block E, Noida
5 yrs of
Exp
4 students
500
per hour
Online Classes
Student's Home
Tutor's home
Dear Parents and students, I am Amit, We provide home-based tuition for subjects and we also do professional coaching for NEET/IIT-JEE and other competitive exams. We have experienced tutors in all the fields and subjects. We also provide demo class free of cost. We do online as well as offline mode of teaching...
Top Tutor
Sun City Vistaar, Bareilly
10 yrs of
Exp
6 students
139 hrs of classes completed
500
per hour
Online Classes
I am teaching since 20011 of class 12th and 12 students. My subject of teaching is mathematics CBSC board.
Superb teaching with basic concept and core theory. His way of explanation is very nice. He teaches basics of mathematics very well.
Other Subjects in CBSE - Class 11
Download free CBSE - Class 11 Mathematics Sets Miscellaneous Exercise on Chapter 1 worksheets
Download Now
