Constructing Proofs in Mathematics: Where do I Begin?

In summary: A\cup B because A\cup B does not include x.The first case is easy. Since x is in A, x is in B\cup A.The second case is a little more complicated. Here, x is in B but not in A\cup B. So, we need to show that x is in A\cup B. To do that, we use the definition of inclusion. Since x is in A\cup B, x is in A. Since x is in A, x is in B\cup A. Since x is in A\cup B, x is in A\cup B. So, x is
  • #1
Repetit
128
2
Hey
Im trying to study abstract algebra, set theory and group theory, on my own. I have trouble understanding how to construct mathematical proofs though. Some of the things the excercises tells me to prove, seems so intuitively clear and obvious that I don't know what's left to prove. For example, prove that

[tex]
A\cup B = B\cup A
[/tex]

where A and B are two sets, or

[tex]
A\cap(B\cup C)=(A\cap B)\cup (A\cap C)
[/tex]

I have no idea how to start. Can someone give me a hint on these? And maybe a hint on general proof making in mathematics?
 
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  • #2
Well, for starters, when you are trying to prove that the the union of A and B is equal to the union of B and A it is necessary to show that the union of A and B is contained in the union of B and A. Then you need to prove the converse. When I do proofs of this sort, I start by drawing some pretty Venn diagrams. It helps me visualize what I am trying to prove.
 
  • #3
The first axiom of elementary set-theory is the Axiom of extensionality:
S=T if and only if ([itex]x \in S[/itex] if and only if [itex]x \in T[/itex]).​
In fact, this is the only axiom of Zermelo set theory that says anything about equality. So, if you want to prove two sets are equal, essentially the only method available is to apply this axiom.

Once you learn more theorems (such as the two theorems you posted), you will learn more ways to prove sets are equal.
 
  • #4
Your book probably defines equality of sets. That definition may or may not be the same as the one Hurkyl gave.

What's the definition given?
 
  • #5
Thanks for the answers.

I think the definition the book gives of equality between sets, is that for two sets A and B to be equal, A must be a subset of B and B must be a subset of A.
 
  • #6
Yes, that's exactly what Hurkyl gave.

And to prove "A is a subset of B" you start "let x be a member of A" and then, using the fact that x satisfies whatever the definition of A is, show that it must also satisfy the definition of B and so "x is a member of B".

Here, since you are asked to prove [itex]A\cup B = B\cup A[/itex] you need to prove that [itex]A\cup B[/itex] is a subset of [itex]B\cup A[/itex] and then prove that [itex]B\cup A[/itex] is a subset of [itex]A\cup B[/itex].

To prove that [itex]A\cup B[/itex] is a subset of [itex]B\cup A[/itex], start "let x be a member of [itex]A\cup B[/itex]. I said above "using the fact that x satisfies whatever the definition of A is". Here the set is defined as a union so we really need to use the definition of "union". x is in [itex]A\cup B[/itex] if and only if x is in A or x is in B. Since that is an "or", break this into two cases:
(i) x is in A. In that case x is in [itex]B\cup A[/itex] because ...
(ii) x is in B. In that case x is in [itex]B\cup A[/itex] because ...
 

1. What is a mathematical proof?

A mathematical proof is a logical argument that uses established axioms, definitions, and previously proven theorems to demonstrate the truth of a mathematical statement.

2. Why are proofs important in mathematics?

Proofs are essential in mathematics because they provide a rigorous and systematic way to verify the validity of mathematical statements. They allow for the discovery of new theorems and provide a solid foundation for further mathematical exploration.

3. What are the different types of proofs?

There are several types of proofs in mathematics, including direct proofs, indirect proofs, proof by contradiction, proof by induction, and proof by contrapositive. Each type of proof has its own unique approach and is suitable for proving different types of statements.

4. How do mathematicians know when a proof is valid?

Mathematicians use a set of rules and techniques known as proof techniques to evaluate the validity of a proof. These techniques include logical reasoning, the use of symbols and notation, and the application of mathematical concepts and properties.

5. Can a proof be wrong?

Yes, a proof can be wrong if there is an error in logic or if the assumptions used are incorrect. It is important for mathematicians to carefully check and verify their proofs before considering them valid.

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