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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 (Arts) 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

To Show: A cup B

Clearly, B subset A cup B

Let x in A cupB

=>x in A or x inB

Case I : x in A

=>x in B        [because AsubsetB]

therefore A cup B subset B

Case II : x in B

Then,  A cup B = B

Conversely, let A cup B = B

Let x ∈ A

=> x in A cup B    [because A subset A cup B]

=> x in B             [because A cup B = B]

therefore A subset B

Hence, (i) ⇔ (iii)

Now, we have to show that (i) ⇔ (iv).

Let A ⊂ B

Clearly A nn B subset A

Let ∈ A

We have to show that x in A nn B

As A ⊂ B, ∈ B

therefore x in A nn B

therefore A subset A nn B

Hence, A = A ∩ B

Conversely, suppose A ∩ B = A

Let x ∈ A

⇒ x ∈ A ∩ B

⇒ x ∈ A and x ∈ B

⇒ ∈ B

∴ A ⊂ B

Hence, (i) ⇔ (iv).

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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 Concept: Subsets.
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