Can someone help me prove that: ((A cap B)⊕A)′=A′ cup B.

Noelanijd 2022-07-15 Answered
Can someone help me prove that:
( ( A B ) A ) = A B
A′ is A complement.
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Answers (2)

berouweek
Answered 2022-07-16 Author has 11 answers
Step 1
aka Xor p q ( p ¬ q ) ( ¬ p q ) ( p q ) ( ¬ p ¬ q )
This set notation ' ' corresponding to xor in logic.
Any x ( ( A B ) A ) if and only if:
¬ ( ( x A x B ) x A )
Since ¬ ( ( p q ) p ) ( ¬ p q ) is a tautology,
It's clearly equivalent to x A x B, hence proved…
Apply def. of Xor:
¬ ( ( ( x A x B ) x A ) ( ¬ ( x A x B ) x A ) )
Apply Commutative law & Associative law:
¬ ( ( ( x A x A ) x B ) ( ¬ ( x A x B ) x A ) )
Apply Negation law:
¬ ( ( x B ) ( ¬ ( x A x B ) x A ) )
Apply Domination law:
¬ ( ( ¬ ( x A x B ) x A ) )
Apply Identity law:
¬ ( ¬ ( x A x B ) x A )
Apply De Morgan's law:
¬ ( ( x A x B ) x A )
Apply Distributive law:
¬ ( ( x A x A ) ( x B x A ) )
Apply Negation law:
¬ ( ( x B x A ) )
Apply Domination law:
¬ ( x B x A )
Apply De Morgan's law:
x B x A
Apply Commutative law:
x A x B
This hold if and only if x A B
Step 2
Hence we proved ( ( A B ) A ) = A B
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iarc6io
Answered 2022-07-17 Author has 2 answers
Step 1
Note that denotes the symmetric difference between two sets. In other words,
P Q = ( P Q ) ( Q P ) = ( P Q ) ( Q P )
Step 2
Using this definition as well as set identities, we have the following proof:
( ( A B ) A )
= [ ( ( A B ) A ) ( A ( A B ) ) ] ----- by definition of
= [ ( ( B A ) A ) ( A ( A B ) ) ] ----- by the commutative law
= [ ( B ( A A ) ) ( A ( A B ) ) ] ----- by the associative law
= [ ( B ) ( A ( A B ) ) ] ----- by the complement law
= [ ( A ( A B ) ) ] ----- by the domination law
= [ A ( A B ) ] ----- by the identity law
= A ( A B ) ----- by DeMorgan's law
= A ( A B ) ----- by the double complement law
= ( A A ) ( A B ) ----- by the distributive law
= U ( A B ) ----- by the complement law
= A B ----- by the identity law
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