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Proving De Morgan's Laws

  • Thread starter 385sk117
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  • #1
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Homework Statement



Prove De Morgan's Laws (only (A U B)' = A' n B' part)

Homework Equations



(A U B)' = A' n B'

The Attempt at a Solution



I used this
x does not belong to (A U B)
x belongs to A' and B'
x belongs to A' n B'

now opposite

x belongs to A' n B'
x belogns to A' and B'
x does not belong to A and B

and that's it.
how can this x does not belong to A and B can be changed into
x does not belong to (A U B)' ?

all these proving things are confusing..they are too obvious and that's why i can not do this properly how silly..
Any tips that can be used when solving this kind of questions?

Thankyou!
 

Answers and Replies

  • #2
dx
Homework Helper
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[tex] (x \in A' \cap B') \Rightarrow \neg(x \in A)\wedge \neg (x \in B) = \neg[ (x \in A) \vee (x \in B) ] \Rightarrow (x \in (A \cup B)') [/tex]

Does that make sense?
 
  • #3
311
1
@dx - you are using demorgans law to prove demorgans law....

You used that form of the law that applies to mathematical logic, to get the same law in a different form in set theory..

So that proof is void
 
  • #4
311
1
@dx - you are using demorgans law to prove demorgans law....

You used that form of the law that applies to mathematical logic, to get the same law in a different form in set theory..

So that proof is void
 
  • #5
dx
Homework Helper
Gold Member
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18
How can you prove anything if you're not allowed to use logic?
 
  • #6
311
1
you can use logic. What i was talking about was a branch of mathematics - mathematical logic. You have to start from the axioms and prove the law that u have used in your proof. Start from the axioms of 'Mathematical Logic', and use those to prove it. It can be proved, but u must do it. Once that is done, rest follows

What you have done is use DeMorgan's Law in ML to prove the same in set theory. That is where u are wrong.
 
  • #7
311
1
you can use logic. What i was talking about was a branch of mathematics - mathematical logic. You have to start from the axioms and prove the law that u have used in your proof. Start from the axioms of 'Mathematical Logic', and use those to prove it. It can be proved, but u must do it. Once that is done, rest follows

What you have done is use DeMorgan's Law in ML to prove the same in set theory. That is where u are wrong.
 
  • #8
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what is this \neg stand for?
 
  • #9
HallsofIvy
Science Advisor
Homework Helper
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Homework Statement



Prove De Morgan's Laws (only (A U B)' = A' n B' part)

Homework Equations



(A U B)' = A' n B'

The Attempt at a Solution



I used this
x does not belong to (A U B)
x belongs to A' and B'
x belongs to A' n B'
Very good but more precisely: "Let x belong to (A U B)'. Then x does not belong to A U B and so is not in A and is not in B. Therefore x is in A' and x is in B'. Then x is in A' n B'.

now opposite

x belongs to A' n B'
x belogns to A' and B'
x does not belong to A and B
Let x be in A' n B'. Then x is in A' and in B'. Since x is in A', x is not in A. Since x is in B', x is not in B. If x were in A U B, it would have to be in either A or B. Since it is not, it is not in A U B and so is in A' U B'.

and that's it.
how can this x does not belong to A and B can be changed into
x does not belong to (A U B)' ?

all these proving things are confusing..they are too obvious and that's why i can not do this properly how silly..
Any tips that can be used when solving this kind of questions?

Thankyou!
 
Last edited by a moderator:

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