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
1.2k views
in Mathematics by (8.0k points)

For all sets A, B and C

Is (A – B) ∩ (C – B) = (A ∩ C) – B?

Justify your answer.

1 Answer

0 votes
by (13.2k points)
selected by
 
Best answer

Yes

Let x ∈ (A – B) ∩ (C – B)

⇒ x ∈ A – B and x ∈ C – B

⇒ (x ∈ A and x ∉ B) and (x ∈ C and x ∉ B)

⇒ (x ∈ A and x ∈ C) and x ∉ B

⇒ (x ∈ A ∩ C) and x ∉ B

⇒ x ∈ (A ∩ C) – B

So,

(A – B) ∩ (C – B) ⊂ (A ∩ C) – B    ....(1)

Now, conversely

Let 

y ∈ (A ∩ C) – B

⇒ y ∈ (A ∩ C) and y ∉ B

⇒ (y ∈ A and y ∈ C) and (y ∉ B)

⇒ (y ∈ A and y ∉ B) and (y ∈ C and y ∉ B)

⇒ y ∈ (A – B) and y ∈ (C – B)

⇒ y ∈ (A – B) ∩ (C – B)

So (A ∩ C) – B ⊂ (A – B) ∩ (C – B) ... (2)

From (1) and (2), 

(A – B) ∩ (C – B) = (A ∩ C) – B

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

...