For any two sets- prove that -i A -A - B-Aii A -A - B-A

(i) A ∪ (A ∩ B)= (A ∪ A) ∩ (A ∪ B) [ ∴ union ∪ is distributive overintersection ∩] = A ∩ (A ∪ B) [∴ A ∪ A = A] = A [∴ A ⊂ (A ∪ B), as union of two ...

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Standard XII

Mathematics

Intersection

For any two s...

Question

For any two sets, prove that :

(i) $A \cup (A \cap B) = A$

(ii) $A \cap (A \cup B) = A$

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Solution

(i) $A \cup (A \cap B) = (A \cup A) \cap (A \cup B)$

[ $∴$ union $\cup$ is distributive overintersection $\cap$ ]

= $A \cap (A \cup B)$ [ $∴ A \cup A = A$ ]

= A [ $∴ A \subset (A \cup B)$ , as union of two sets is bigger than each of the individual sets]

Hence , $A \cup (A \cap B) = A$ Proved.

(ii) $A \cap (A \cup B) = (A \cap A) \cup (A \cap B)$

[ $∵ \cap$ distributes over $\cup$ ]

= $A \cup (A \cap B) [∵ A \cap A = A]$

= A [using (i)]

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For any two sets, prove that : (i) A∪(A∩B)=A (ii) A∩(A∪B)=A

For any two sets, prove that :

(i) $A \cup (A \cap B) = A$

(ii) $A \cap (A \cup B) = A$