Natural deduction sets (Rules of nature deduction)

[emanoelvianna]

Hello fine.

I'm studying logic and great difficulties to understand its principles, and should prove some theories involving the laws of identity of sets of mathematics using the method of natural deduction, they are:

a) A ∪ ∅ = A
b) A ∩ ∅ = ∅

I am trying as follows, but I can not solve

http://www.imagesup.net/dm-1514135726215.png

Could anyone help me solve ?!
If I could be pointed out to me some book or website to get more doubts which were to appear on deduction of sets I'd appreciate it.
Thank you.

 

[mathman]

I am not sure what you are trying to do, but the basic logic is that the empty set has no elements, so if it is the union it must be in A.

 

[emanoelvianna]

Hello, thank you by return.

I understand the theory that a set, but I need to prove it by natural deduction, this theory is known as "Natural deduction rules for theory set"

Link to example: http://tellerprimer.ucdavis.edu/pdf/1ch6.pdf

 

[Stephen Tashi]

Link to example: http://tellerprimer.ucdavis.edu/pdf/1ch6.pdf

The usual way to do proofs about set theory identities is to use logic that involves quantifiers, such as "for each" and "there exists". ( symbolized by [itex] \forall [/itex] and [itex] \exists [/itex]). The link you gave is about using the more elementary type of logic that lacks quantifiers.

In the link you gave, [itex] A \supset B[/itex] does not mean that [itex] B [/itex] is a subset of [itex] A [/itex]. In the link, [itex]A \supset B [/itex] means "[itex] A [/itex] implies [itex] B [/itex]". The link you gave is not about set theory.

 

[blue_raver22]

Hello,

Natural deduction is a method of reasoning in logic that is based on the principles of deduction and induction. It is commonly used in mathematics to prove theorems and establish logical relationships between statements. In order to solve the problems you have mentioned, we can use the following rules of natural deduction:

1. Identity Introduction (ID): This rule states that if we have a statement A, we can introduce a new statement A=A. This is used to prove the identity of sets.

2. Identity Elimination (IE): This rule states that if we have a statement A=A, we can eliminate the identity to obtain A. This is used to prove the equality of sets.

3. Union Introduction (UI): This rule states that if we have two statements A and B, we can introduce a new statement A∪B. This is used to prove the union of sets.

4. Union Elimination (UE): This rule states that if we have a statement A∪B, we can eliminate the union to obtain either A or B. This is used to prove that a set is a subset of another set.

5. Intersection Introduction (II): This rule states that if we have two statements A and B, we can introduce a new statement A∩B. This is used to prove the intersection of sets.

6. Intersection Elimination (IE): This rule states that if we have a statement A∩B, we can eliminate the intersection to obtain either A or B. This is used to prove that a set is a subset of another set.

Now, let's solve the problems you have mentioned using these rules:

a) A ∪ ∅ = A
Proof:
1. A∪∅ (Assumption)
2. A (UE 1)

b) A ∩ ∅ = ∅
Proof:
1. A∩∅ (Assumption)
2. A (IE 1)
3. ∅ (UE 1)

I hope this helps you understand the principles of natural deduction and how to apply them to solve problems involving sets. For further resources, I would recommend checking out textbooks on logic and set theory, as well as online resources such as Stanford Encyclopedia of Philosophy and Internet Encyclopedia of Philosophy.

Best of luck in your studies!