Idempotent laws For any set A- prove that-i A - A - Aii A - A - A

We have (i) A∪ A = A{ x:xϵ A or xϵ A }={ x:xϵ A }=A (ii) A ∩ A=A{ x:xϵ A and xϵ A }={ x:xϵ A }=A ...

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Differentiability in an Interval

Idempotent la...

Question

(Idempotent laws) For any set A, prove that:
$(i) A \cup A = A$

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

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Solution

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A \cup (A \cap B) = A

A \cap (A \cup B) = A

For any sets A and B, prove that:
$(i) A \cup (A \cap B) = A$

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

(De Morgan's laws) For any two sets A and B, prove that:
$I . (A \cup B)^{'} = (A^{'} \cap B^{'})$

$I I . (A \cap B)^{'} = (A^{'} \cup B^{'})$

(Commultative laws) For any two sets a and B, prove that:
$I . A \cup B = B \cup A$ [Commutative law for union of sets]

$I I . A \cap B = B \cap A$ [Commutative law for intersection of sets]

(Distributive laws) For any three sets A, B, C prove that:

I. $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
[Distirbutive law of union over intersection]

II. $A \cap [(B \cup C)] = (A \cap B) \cup (A \cap C)$
[Distributive law of intersection over union]

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