Need to check proof of De Morgan's law

• issacnewton
In summary, the conversation is about a proof for the statement A\cap B=A\setminus(A\setminus B). The individual has completed the proof and is asking for verification. They have also mentioned the need to do a reverse proof.
issacnewton

Homework Statement

Hi

I have done the proof, just need to check it. I wanted to prove that

$A\cap B=A\setminus(A\setminus B)$

Proof:

The Attempt at a Solution

$\mbox{let}\ x\in A\cap B\ \mbox{be arbitrary}$
$\Rightarrow x \in A\ \mbox{and}\ x \in B$
$\Rightarrow x \in A\ \mbox{and}\ x \notin B^{c}$
$\Rightarrow x \in A\ \mbox{and}\ x \notin A\setminus B$
$\Rightarrow x \in A\setminus (A\setminus B)$

I know that I have to do reverse proof too. But is it ok so far ?

Thanks

This is correct so far as it goes, yes.

1. What is De Morgan's law?

De Morgan's law is a mathematical theorem that states the relationship between logical operators "and" and "or" when negated. It states that the negation of a conjunction (and) is equivalent to the disjunction (or) of the negations of the individual statements, and vice versa.

2. Why is it important to check the proof of De Morgan's law?

Checking the proof of De Morgan's law is important because it ensures the validity and accuracy of the theorem. It allows us to verify that the law holds true for all possible scenarios and avoid any errors or inconsistencies.

3. How can De Morgan's law be applied in real life situations?

De Morgan's law has applications in logic, set theory, and computer science. In real life situations, it can be used to simplify complex logical expressions and make them easier to understand. It is also used in digital circuit design and programming.

4. Can De Morgan's law be extended to more than two statements?

Yes, De Morgan's law can be extended to more than two statements. This is known as De Morgan's extended law and it states that the negation of a disjunction (or) of multiple statements is equivalent to the conjunction (and) of the negations of each individual statement.

5. Are there any other versions of De Morgan's law?

Yes, there are two other versions of De Morgan's law: one for quantifiers and one for modal logic. The quantifier version states that the negation of a universal quantifier is equivalent to the existential quantifier of the negated statement, and vice versa. The modal logic version is used in modal logic systems and states that the negation of a necessary or possible statement is equivalent to the possibility or necessity of the negated statement, respectively.

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