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If X and Y are subsets of the universal set U, then show that 

(i) Y ⊂ X ∪ Y (ii) X ∩ Y ⊂ X (iii) X ⊂ Y ⇒ X ∩ Y = X

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(i) X ∪ Y = {x | x ∈ X or x ∈ Y} 

Thus x ∈ Y ⇒ x ∈ X ∪ Y 

Hence, Y ⊂ X ∪ Y

(ii) X ∩ Y = {x | x ∈ X and x ∈ Y} 

Thus x ∈ X ∩ Y ⇒ x ∈ X 

Hence X ∩ Y ⊂ X 

(iii) Note that 

x ∈ X ∩ Y ⇒ x ∈ X 

Thus X ∩ Y ⊂ X 

Also, since X ⊂ Y, 

x ∈ X ⇒ x ∈ Y ⇒ x ∈ X ∩ Y 

so that X ⊂ X ∩ Y 

Hence the result X = X ∩ Y follows.

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