# Proving De Morgan's Laws

## 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
Homework Helper
Gold Member
$$(x \in A' \cap B') \Rightarrow \neg(x \in A)\wedge \neg (x \in B) = \neg[ (x \in A) \vee (x \in B) ] \Rightarrow (x \in (A \cup B)')$$

Does that make sense?

@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 - 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
Homework Helper
Gold Member
How can you prove anything if you're not allowed to use logic?

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.

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.

what is this \neg stand for?

HallsofIvy
Homework Helper

## 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!

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