Discrete Math Proof: A cup B. The problem I'm having trouble proving is the following: A cup B=(A cap BC) cup (AC cap B) cup (A cap B) , where C denotes complement of a set

Luciano Webster 2022-07-16 Answered
Discrete Math Proof: A B
The problem I'm having trouble proving is the following:
A B = ( A B C ) ( A C B ) ( A B ), where C denotes complement of a set
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Answers (2)

umgangistbf
Answered 2022-07-17 Author has 12 answers
Step 1
Intuitively the result says that if an element is in A B, then either the element is in A and not B, B and not A, or in both A and B.
Step 2
The arguement can then proceed as follows:
x A B x A  and  x B , or  x A  and  x B , or  x A  and  x B x ( A B c ) ( A c B ) ( A B )

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Ethen Frey
Answered 2022-07-18 Author has 6 answers
Step 1
First, you should not begin your proof by stating what you intend to prove. Instead of starting by writing A B = ( A B C ) ( A C B ) ( A B ) and aiming to reach A B = A B, you should start with ( A B C ) ( A C B ) ( A B ) and show that it equals (through rewriting it in various forms that are the same set) as A B.
(You can also show that two sets are equal by showing that each is a subset of the other; I don’t mean that my suggestion is the only approach, but it’s the appropriate way to write the proof using the particular mathematical argument you are presenting.)
This may seem like a silly question of style, but the fact is that you have only explained that if what you want to prove happens to be true, then it must follow that A B = A B. This is not a proof. It would also follows that A B = A B even if what you began with is false. Any assumptions, even false ones, logically imply a true statement. A conjecture is not proven true by showing that you can use it to conclude a true statement. You have to show that you can proceed logically from true statements to prove it.
Step 2
Second, you should add some parentheses. At one point (after you use the distributive law), you write the expression A ( B C B ) ( A C B ). You intend for this to mean ( A ( B C B ) ) ( A C B ), and it’s safest to say so, because combinations of and may give different results depending on the order in which the operators are evaluated.

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