Proof of a simple logic statement

In summary, the statement "For all x in S, is equivalent to there does not exist an x in S" is false. The correct statement should be "For all x in S, is equivalent to the negation of there exists an x not in S."
  • #1
soopo
225
0

Homework Statement


[tex]\forall x \in S <-> \exists x \not \in S[/tex]

The Attempt at a Solution


The statement is clearly false.
I will try to show that by the proof of contradiction.

Let
[tex]P: \forall x \in S[/tex]
and
[tex]Q: \exists x \not\in S [/tex]

The negation of Q is
[tex]negQ: \forall x \not\in S [/tex]
and the negation of P is
[tex]negP: \exists x \in S [/tex].

negQ means that all x are not in S, while negP means that there is one x in S.
This is a contradiction.

Therefore, the original statement must be false.

Is my proof correct?
 
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  • #2
Your statement is a bit odd.
"For all x in S, is equivalent to there exists an x such that there exists an S"
What is on the right hand side? What does x exist in and there exist x and S such that what?In first-order logic, your statement is not a well-formed formula :tongue:
 
  • #3
CompuChip said:
"For all x in S, is equivalent to there exists an x such that there exists an S"
What is on the right hand side? What does x exist in and there exist x and S such that what?

The statement should be "For all x in S, is equivalent to there does not exist an x in S."

The rest of the proof is also now as I want.
 
  • #4
CompuChip said:
Your statement is a bit odd.
"For all x in S, is equivalent to there exists an x such that there exists an S"
What is on the right hand side? What does x exist in and there exist x and S such that what?


In first-order logic, your statement is not a well-formed formula :tongue:

I agree with you. The statement is still false.

Perhaps, the initial statement should be
[tex] \forall x \in S <=> \neg(\exists x \not \in S) [/tex]

It means that: all x in S is equivalent with the negation of the statement that
there exists x not in S.
 

1. What is a simple logic statement?

A simple logic statement is a sentence or proposition that can either be true or false. It is composed of variables, logical operators, and quantifiers, and it follows rules of logical reasoning.

2. What is a proof of a simple logic statement?

A proof of a simple logic statement is a step-by-step demonstration of how the statement can be logically derived from a set of axioms or assumptions. It aims to show that the statement is true based on the rules of logic.

3. How do you construct a proof of a simple logic statement?

To construct a proof of a simple logic statement, you need to first identify the logical operators and variables present in the statement. Then, you can use logical rules and laws to manipulate the statement and arrive at a logical conclusion. It is important to justify each step in the proof using previously established logical rules.

4. What is the purpose of a proof in logic?

The purpose of a proof in logic is to provide a rigorous and systematic method of verifying the truth of a statement. It allows us to confidently accept a statement as true based on logical reasoning rather than intuition or belief.

5. Can a proof of a simple logic statement be wrong?

Yes, a proof of a simple logic statement can be wrong if it contains a logical error or if the initial assumptions or axioms were incorrect. It is important to carefully check each step in a proof to ensure its validity.

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