Show that-i A - A - B - Aii To show A - A - B - A

(i) A ⊂ A (a is a sbuset of if ) property of sets amp; A ∩ B ⊂ A (naturel) ∴ A ∪ (A ∩ B)⊂ A (1) amp; A ⊂ A ∪ (A ∩ B) (2) From (1) amp; (2) A ∪ (A ∩ B ...

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Standard XII

Mathematics

De Morgan's Law

Show that:i A...

Question

Show that:
(i) $A \cup (A \cap B) = A$
(ii) To show $A \cap (A \cup B) = A$

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Solution

(i) $A \subset A$ (a is a sbuset of if )-property of sets & $A \cap B \subset A$ (naturel)
$∴ A \cup (A \cap B) \subset A - (1)$
& $A \subset A \cup (A \cap B) - (2)$
From (1) & (2) $A \cup (A \cap B) = A$
(ii) $A \cap (A \cup B) = (A \cap A) \cup (A \cap B)$ -property of sets.
$= A \cup (A \cap B)$
=A {from (i)above}
$∴$ proved.

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Q. Using properties of sets show that:

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Q. For any two sets, prove that:
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Show that: (i) A∪(A∩B)=A (ii) To show A∩(A∪B)=A

(i) A⊂A (a is a sbuset of if )-property of sets & A∩B⊂A (naturel) ∴A∪(A∩B)⊂A−(1) & A⊂A∪(A∩B)−(2) From (1) & (2) A∪(A∩B)=A (ii) A∩(A∪B)=(A∩A)∪(A∩B)-property of sets. =A∪(A∩B) =A {from (i)above} ∴ proved.

Show that:
(i) $A \cup (A \cap B) = A$
(ii) To show $A \cap (A \cup B) = A$