Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
177 views
in Sets, Relations and Functions by (42.5k points)
closed by

For any sets A and B, prove that

(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A)

1 Answer

+1 vote
by (44.7k points)
selected by
 
Best answer

Given: A and B two sets are given. 

Need to prove: (A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A) 

Let us consider, (x, y)∈(A × B) ∩ (B × A) 

⇒ (x, y)∈(A × B) and (x, y)∈(B × A) 

⇒ (x∈A and y∈B) and (x∈B and y∈A)

⇒ (x∈A and x∈B) and (y∈B and y∈A) 

⇒ x∈(A ×B) and y∈(B × A)

⇒ (x, y)∈(A × B) ∩ (B × A) 

From this, we can conclude that, 

⇒ (A × B) ∩ (B × A) ⊆ (A ∩ B) × (B ∩ A)..... (1) 

Let us consider again, (a, b)∈(A ∩ B) × (B ∩ A) 

⇒ a∈(A ∩ B) and b∈(B ∩ A) 

⇒ (a∈A and a∈B) and (b∈B and b∈A) 

⇒ (a∈A and b∈B) and (a∈B and b∈A) 

⇒ (a, b)∈(A × B) and (a, b)∈(B × A) 

⇒ (a, b)∈(A × B) ∩ (B × A) 

From this, we can conclude that, 

⇒ (A ∩ B) × (B ∩ A) ⊆ (A × B) ∩ (B × A) ...... (2) 

Now by the definition of set we can say that, from (1) and (2), 

(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A) [Proved]

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...