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