CBSE (Science) Class 11CBSE
Account
It's free!

User


Login
Create free account


      Forgot password?
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Solution - Show that the Following Four Conditions Are Equivalent:A Subset B, a Difference B = φ, a Union B = B And a Intersection B = a - CBSE (Science) Class 11 - Mathematics

Question

Show that the following four conditions are equivalent:

(i) A ⊂ B

(ii) A – B = Φ

(iii) A ∪ B = B 

(iv) A ∩ B = A

Solution

First, we have to show that (i) ⇔ (ii).

Let A ⊂ B

To show: A – B ≠ Φ

If possible, suppose A – B ≠ Φ

This means that there exists x ∈ A, x ≠ B, which is not possible as A ⊂ B.

∴ A – B = Φ

∴ A ⊂ B ⇒ A – B = Φ

Let A – B = Φ

To show: A ⊂ B

Let x ∈ A

Clearly, ∈ B because if x ∉ B, then A – B ≠ Φ

∴ A – B = Φ ⇒ A ⊂ B

∴ (i) ⇔ (ii)

Let A ⊂ B

Conversely, suppose A ∩ B = A

Let x ∈ A

⇒ x ∈ A ∩ B

⇒ x ∈ A and x ∈ B

⇒ ∈ B

∴ A ⊂ B

Hence, (i) ⇔ (iv).

Is there an error in this question or solution?

APPEARS IN

Reference Material

Solution for question: Show that the Following Four Conditions Are Equivalent:A Subset B, a Difference B = φ, a Union B = B And a Intersection B = a concept: Subsets. For the courses CBSE (Science), CBSE (Arts), CBSE (Commerce)
S