Question

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

Answer 1

Given :  X and Y are subsets of the universal set U,

To Find :  then show that

(i) Y ⊂ X ∪ Y

(ii) X ∩ Y ⊂ X  

(iii) X ⊂ Y ⇒ X ∩ Y = X

Solution:

(i) X ∪ Y = {x | x ∈ X or x ∈ Y}

  x ∈ Y ⇒ x ∈ X ∪ Y

  Y ⊂ X ∪ Y

(ii) X ∩ Y = {x | x ∈ X and x ∈ Y}

x ∈ X ∩ Y ⇒ x ∈ X

Hence X ∩ Y ⊂ X

(iii)  

x ∈ X ∩ Y ⇒ x ∈ X

X ∩ Y ⊂ X

X ⊂ Y,

x ∈ X ⇒ x ∈ Y

⇒ x ∈ X ∩ Y

X = X ∩ Y

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