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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

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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).

  Is there an error in this question or solution?

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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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