Math, asked by ramadevi704216, 7 months ago

the necessary and sufficient condition for a set Y to be a subset of X is that X U Y=X​

Answers

Answered by ArunSivaPrakash
0

Given:

The necessary and sufficient condition for a set Y to be a subset of X is that X U Y = X.

To Find:

We have to prove that "​the necessary and sufficient condition for a set Y to be a subset of X is that X U Y = X​."

Solution:

Let Y ⊂ X, then

x ∈ Y ⇒ x ∈ X.

Now, x ∈ X ∪ Y ⇒ x ∈ X or x ∈ Y.

⇒ x ∈ X

∴ X ∪ Y ⊂ X.

Also,  X ⊂ X ∪ Y.

From above equations, X ∪ Y = X.

Conversely, if X ∪ Y = X, we have to prove that Y ⊂ X.

Now, X ∪ Y = X ⇒ X ∪ Y ⊂ X and X ⊂ X ∪ Y.

⇒ X ∪ Y ⊂ X

i.e., X ⊂ X and Y ⊂ X

⇒ Y ⊂ X.

Hence it is proved that "​the necessary and sufficient condition for a set Y to be a subset of X is that X U Y = X​."

#SPJ1

Answered by arshikhan8123
3

Concept-

A sufficient condition guarantees the truth of another condition, but is not necessary for that condition to occur. A necessary condition is required for something else to happen, but it does not guarantee that something else will happen.

Given-

Necessary and sufficient condition for set Y to be the subset of X is that (X) U (Y) = X.

Find-

Prove that "​the necessary and sufficient condition for  set Y to be the subset of X is that (X) U (Y) = X​."

Solution-

Let Y ⊂ X, then

x ∈ Y ⇒ x ∈ X.

Now, x ∈ (X ∪ Y) ⇒ x ∈ X or x ∈ Y.

⇒ x ∈ X

∴ (X ∪ Y) ⊂ X.

Also,  X ⊂ (X ∪ Y).

From above equations, (X ∪ Y) = X.

Conversely, if (X ∪ Y) = X, we have to prove that Y ⊂ X.

Now, (X ∪ Y) = X ⇒ (X ∪ Y) ⊂ X and X ⊂ (X ∪ Y).

⇒ (X ∪ Y) ⊂ X

i.e., X ⊂ X and Y ⊂ X

⇒ Y ⊂ X.

Therefore,  "​the necessary and sufficient condition for a set Y to be a subset of X is that (X) U (Y) = X​."

#SPJ1

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