How Do You Prove De Morgan's Law for Union and Complement?

[385sk117]

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!

 

[dx]

$$(x \in A' \cap B') \Rightarrow \neg(x \in A)\wedge \neg (x \in B) = \neg[ (x \in A) \vee (x \in B) ] \Rightarrow (x \in (A \cup B)')$$

Does that make sense?

 

[praharmitra]

@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

 

[praharmitra]

@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

 

[dx]

How can you prove anything if you're not allowed to use logic?

 

[praharmitra]

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.

 

[praharmitra]

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.

 

[385sk117]

what is this \neg stand for?

 

[HallsofIvy]

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!