Proving 2 collections are equivalent making use of bi - directional set incorporation

Prove $(A∩B)’=A’∪B’$.
Allow $x ∈ (A ∩ B)’$
$∴ (x ∈ A ∩ B)’$
$∴ (x ∈ A ∧ x ∈ B)’$
$∴ (x ∈ A)’ ∨ (x ∈ B)’$
$∴ x ∈ A’ ∨ x ∈ B’$
$∴ x ∈ A’ ∪ B’$
$∴ (A ∩ B)’ ⊆ A’ ∪ B’$

The above is the remedy given. Sorry if it appears unimportant yet I do not recognize just how
$∴ (x ∈ A ∧ x ∈ B)’$
brings about
$∴ (x ∈ A)’ ∨ (x ∈ B)’$
and also not
$∴ (x ∈ A)’ ∧ (x ∈ B)’$

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2019-05-19 00:04:26
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Answers: 1

This is simply among DeMorgan is Laws. Not (An and also B) amounts (Not A) or (Not B), that is that (An and also B) is incorrect if either one is incorrect. If you most likely to (Not A) and also (Not B) you require both to be incorrect.

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2019-05-21 12:22:25
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