Started learning proofs - need some feedback

In summary: For example, both Cantor and Peano had set theories, but neither one quite had what we now consider ZF. Cantor's was too naive and Peano's didn't quite have extensionality. The first person to have what we would recognize as ZF was someone who wrote in 1908, about 20 years after both Cantor and Peano wrote. (Though there were a few people in between who had set theories, such as Dedekind, Frege, Whitehead and Russell, and Zermelo.)In summary, the conversation was about learning advanced mathematics and writing proofs. The proposition being discussed was about proving that if a family of sets contains a particular set, then the intersection
  • #1
4Fun
10
0
Hello guys,

this is my first post on this forum. I want to learn advanced/pure mathematics basically just because I find it really interesting and challenging and I have started to learn about proofs. I'm currently reading Velleman's book and I have reached the part in which you actually start to learn writing proofs. Since Velleman only offers solution for some of the proofs I don't know whether my proofs are actually valid. I would really appreciate if someone would be willing to quickly take a look at some proofs I write and give me some feedback.

Proposition: Proove that if F is a family of sets and A [itex]\in[/itex] F, then [itex]\cap[/itex] F [itex]\subseteq[/itex] A.

Ok I'll start with my scratch work:

Givens: A [itex]\in[/itex] F
Goal: [itex]\cap[/itex] F [itex]\subseteq[/itex] A

[itex]\cap[/itex] F [itex]\subseteq[/itex] A is equivalent to [itex]\forall[/itex] x (x [itex]\in[/itex] [itex]\cap[/itex] F -> x [itex]\in[/itex] A).

Now I let x be an arbitrary element.

Question here: Does x have to be an element or a set? Because [itex]\cap[/itex]F consists only of sets right?!

Then I assume that x [itex]\in[/itex] [itex]\cap[/itex] F.

Givens: A [itex]\in[/itex] F, x [itex]\in[/itex] [itex]\cap[/itex] F
Goal: x [itex]\in[/itex] A

Now x [itex]\in[/itex] [itex]\cap[/itex] F means that [itex]\forall[/itex] A [itex]\in[/itex] F (x [itex]\in[/itex] A) for some A.

So basically that for every element ( or set of F, since F is a family of sets) x is an element of that set. Since A [itex]\in[/itex] F, x is also an element of A.

Now the formal proof:

Let x be arbitrary. Suppose that x [itex]\in[/itex] [itex]\cap[/itex] F, which means that for all sets of F, x is an element of each of those sets. Since A is one of those sets, it follows that x is an element of A. Since x was arbitrary it follows that in general if A [itex]\in[/itex] F then [itex]\cap[/itex] F [itex]\subseteq[/itex] A.

Now although I think that my scratch work was correct, I think the formal proof still sounds incorrect. Could anybody please give my some feedback?
 
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  • #2
I just started to learn proofs as well so take my comments with a grain of salt and wait for the official mentors/helpers before jumping to conclusions , but everything in mathematic is a set.So if ##F=\{A,B,C\}## , then ##F## is simply a set with multiples sets as it's elements so ##A \in F## , ##B \in F## and ##C \in F##.

So ##A## , ##B## , ##C## are elements of ##F## , but that doesn't mean that they aren't sets as well.

Suppose ##A=\{1,2\}## , ##B=\{1,3\}## and ##C=\{1,4\}##.Then ##\bigcap F = \{1\}## and it's true that ##\{1\} \subseteq A##.The logic will work even if there's no intersection , like for exemple if ##A=\{1\}## , ##B = \{2\}## and ##C = \{3\}##.Then ##\bigcap F = \varnothing##.This is the empty set.You can safely assume that the empty set is a subset of ALL sets.So ##\varnothing \subseteq A##.
 
Last edited:
  • #3
4Fun said:
Hello guys,

this is my first post on this forum. I want to learn advanced/pure mathematics basically just because I find it really interesting and challenging and I have started to learn about proofs. I'm currently reading Velleman's book and I have reached the part in which you actually start to learn writing proofs. Since Velleman only offers solution for some of the proofs I don't know whether my proofs are actually valid. I would really appreciate if someone would be willing to quickly take a look at some proofs I write and give me some feedback.

Proposition: Proove that if F is a family of sets and A [itex]\in[/itex] F, then [itex]\cap[/itex] F [itex]\subseteq[/itex] A.

Ok I'll start with my scratch work:

Givens: A [itex]\in[/itex] F
Goal: [itex]\cap[/itex] F [itex]\subseteq[/itex] A

[itex]\cap[/itex] F [itex]\subseteq[/itex] A is equivalent to [itex]\forall[/itex] x (x [itex]\in[/itex] [itex]\cap[/itex] F -> x [itex]\in[/itex] A).

Now I let x be an arbitrary element.

Question here: Does x have to be an element or a set? Because [itex]\cap[/itex]F consists only of sets right?!

Then I assume that x [itex]\in[/itex] [itex]\cap[/itex] F.

Givens: A [itex]\in[/itex] F, x [itex]\in[/itex] [itex]\cap[/itex] F
Goal: x [itex]\in[/itex] A

Now x [itex]\in[/itex] [itex]\cap[/itex] F means that [itex]\forall[/itex] A [itex]\in[/itex] F (x [itex]\in[/itex] A) for some A.

So basically that for every element ( or set of F, since F is a family of sets) x is an element of that set. Since A [itex]\in[/itex] F, x is also an element of A.

Now the formal proof:

Let x be arbitrary. Suppose that x [itex]\in[/itex] [itex]\cap[/itex] F, which means that for all sets of F, x is an element of each of those sets. Since A is one of those sets, it follows that x is an element of A. Since x was arbitrary it follows that in general if A [itex]\in[/itex] F then [itex]\cap[/itex] F [itex]\subseteq[/itex] A.

Now although I think that my scratch work was correct, I think the formal proof still sounds incorrect. Could anybody please give my some feedback?

That seems alright. The formal proof is close to something I would write.

Question here: Does x have to be an element or a set? Because [itex]\cap[/itex]F consists only of sets right?!

What you did was: let ##x## be arbitrary. Assume that ##x\in \bigcap F##. That is alright, but it doesn't tell us what ##x## is. So you might call it a bit vague. It might be better to contract this into one sentence: "Let ##x## be an arbitrary element of ##\bigcap F##."
 
  • #4
reenmachine said:
I just started to learn proofs as well so take my comments with a grain of salt and wait for the official mentors/helpers before jumping to conclusions , but everything in mathematic is a set.

Actually, there are variants of set theory that have classes as well as sets, and variants that have so-called urelements or individuals that are not sets but can be members of sets. This allows one to talk of things like sets of elephants or sets of apples. There is even a variant with individuals and a universal set, called NFU.

I find the history of modern ideas pretty interesting.
 
  • #5



Hello there,

First of all, congratulations on starting to learn about proofs! It can be a challenging but rewarding journey.

In terms of your scratch work, it looks like you have a good understanding of the concept and the goal you are trying to prove. In general, when writing a proof, it is important to clearly state your assumptions and goals, as you have done. However, there are a few things that could be improved in your formal proof:

1. The statement "Now x \in \cap F means that \forall A \in F (x \in A) for some A" is not entirely accurate. The correct statement should be "For all A \in F, x \in A." This means that for any set A in the family F, x is an element of A. The "for some A" part is not necessary.

2. In your formal proof, you start with "Let x be arbitrary," which is a good start. However, you then use the phrase "Suppose that x \in \cap F," which is not necessary. Since you are starting with an arbitrary x, you can simply say "For all A \in F, x \in A," as you did in your scratch work.

3. In your formal proof, you say "Since A is one of those sets," but it is not clear what "those sets" refer to. It would be clearer to say "Since A is an arbitrary set in F."

Overall, your understanding of the concept is correct, but there are some minor errors in your formal proof. I would suggest practicing more proofs and seeking feedback from others to improve your skills. Good luck with your studies!
 

1. What are proofs in science and why are they important?

A proof in science is a logical argument or evidence that supports a scientific claim or theory. Proofs help to verify the validity of scientific knowledge and provide a solid foundation for further research and experimentation.

2. How do you construct a scientific proof?

To construct a scientific proof, you first need to clearly define your hypothesis or claim. Then, gather relevant data and evidence through experiments or observations. Finally, analyze and interpret the data to support your hypothesis using logical reasoning and scientific principles.

3. What are the common mistakes to avoid when writing a scientific proof?

Some common mistakes to avoid when writing a scientific proof include using faulty or biased data, making assumptions without sufficient evidence, and failing to consider alternative explanations. It is also important to clearly explain each step of your reasoning and avoid using jargon or complex language that may be difficult for others to understand.

4. How can I improve my skills in writing scientific proofs?

The best way to improve your skills in writing scientific proofs is through practice and getting feedback from others. Additionally, staying updated on current scientific research and understanding the principles of logic and critical thinking can also help you to write more effective proofs.

5. Are there different types of proofs in science?

Yes, there are several types of proofs in science, including deductive, inductive, and abductive reasoning. Deductive reasoning involves drawing conclusions from general principles, while inductive reasoning involves making generalizations based on specific observations. Abductive reasoning involves using the most likely explanation to support a claim.

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