Question

The necessary and sufficient conditions for a set Y to be a Subset of X is that.X U Y =x

Answer 1

Answer:

Yes its true that if Y is a subset of X, then X U Y = X.

Answer 2

Answer:

To prove that X U Y = X

Let y be the proper subset of X

SO, Y ⊂ X, and let any rational number be x

so, if x ∈ Y

it means x ∈ X too.

Therefore, x ∈ X U Y gives x ∈ X or x ∈ Y

or, x ∈ X

So, the result is : X U Y ⊂ X    …..equation(i)

We also know :

X ⊂ X U Y     …..equation(ii)

from equation(i) and equation(ii), we get :

X U Y = X

and solving it conversely, we get :

if X U Y = X we prove that Y ⊂ X

So, X U Y = X is same as X U Y ⊂ X and X ⊂ X U Y

or, X U Y ⊂ X

or, X ⊂ X and Y ⊂ X

which gives, Y ⊂ X

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