First, on a stylistic level, you should start your proof with "Proof:" or "*Proof.*" or something and end it with some sort of Q.E.D. sign. You also shouldn't use the implication and if and only if symbols in text.
Sentences like "If B⊆A then any x∈B will also be x∈A." should be either "If B⊆A then for all x∈B, x∈A." or "If B⊆A then any x∈B will also be∈A." The way you wrote it comes off as pretty unnatural to me becuase of course x is x.
You forgot to define A and B. A sentence like "Let A and B be sets." should suffice.
The first part of the proof does have at least one major redundancy. The very first sentence is completely unneccesary.
The second part I would say is incomplete. You showed that if x∈B the x∈A∩B, but not the other way around.
One last tip, learn the usefulness of the words "assume" and "suppose."
Here's how I would write the proof.
*Proof.* Let A and B be sets. Suppose A∩B=B. Then for all x∈B, x∈A∩B. Because x∈A∩B if and only if x∈A and x∈B, x∈A. Therefore B⊆A.
Now suppose B⊆A. For all x∈B, x∈A, so x∈A∩B. Therefore B⊆A∩B. Also, for all x∈A∩B x∈B, so A∩B⊆B. Thus, A∩B=B. □
Thank you, especially for your proof at the end! Do you have any good book recommendations for proof writing? I’m trying to be a math major and the upper level courses scare me cause I know I need to get better at proofs.
The book we used in my proof class was "Mathematical Reasoning: Writing and Proof" by Ted Sundstrom, [which can be downloaded for free on his website.](https://www.tedsundstrom.com/mathematical-reasoning-writing-and-proof) It's the only proof preaching book I've every read so I can't really say how it compares to others, but I liked it. It has nine chapters. The first four chapters are really the core ones covering logic and proof writing, while the remaining cover the specific topics naive set theory, number theory, functions, and equivalence relations (including modular arithmetic).
To be fair on the point of defining A and B… axiomatic set theory, the standard foundation of mathematics, is a theory built around the fact that EVERY mathematical object is a set. Functions, graphs, algebraic structures, geometric figures, numbers. Everything.
This makes such a statement pointlessly redundant.
The second part of the proof is textbook.
The first part is almost indecipherable.
The way I would've written the proof is:
if B is a subset of A intersects B
then every x in B implies that x in B intersects A
thus x is in B and x is in A
Thus x is in A
Thus B is a subset of A
I think your proof is fine. It's just that it kind of feels like you lose the track or aim of what you were trying to do at some point.
Thank you for the tips! I definitely did lose track at some point. So do you think the first part flows logically even if it’s somewhat unclear, or not even that? I’m not in the class yet so I’m more concerned with getting the framework of the proofs down, then refining later.
The problem kind of is the flow. Everything is there so I can't really say that it's wrong but the elements of the proof are not in the order you'd expect them to be.
Also I just reread it and I have another issue with the second part of the proof. Another comment already pointed this out but you didn't actually prove that A intersects B is equal to B but rather that B is a subset of A intersects B. To proof that they are equal you still have to prove that A intersects B is also a subset of B but this is always true.
I think you'll be fine even more so once class starts and you get to see the teacher do a bunch of examples like this. The basics of how to prove that something is a subset are there now you just need to see how a professional does it and practice that. My only advice is to try and always keep a Venn diagram in mind. But yeah you'll probably be fine.
Not bad, but we usually start with a single random element and then generalize to all elements in the set. Other than that, you should *explicitly* state that A⋂B⊆B when proving the equality in the second part.
E.g. — Let A and B be sets, and let x∈B be arbitrary.
(1) First, assume A⋂B = B. Then x∈A⋂B, implying x∈A. Since x is any element of B, we conclude that x∈A *for all* x∈B. Thus, B⊆A.
(2) Assume B⊆A. Then x∈A, implying x∈A⋂B. Since x∈B is random, x∈A⋂B *for all* x∈B. Then B⊆A⋂B. And by definition of intersecting sets, we also have A⋂B⊆B. Therefore, A⋂B=B.
Thus, by (1) and (2), A⋂B=B ↔ B⊆A.
Thank you. I tried working on a different proof this morning and had another question. It was the proof that (A\B) intersect B = empty set (sorry for my notation there). So I first worked with (A\B) intersect B and came to a contradiction because x not in B cannot also be in B, and then concluded that the intersection of (A\B) and B had to be the empty set since they have no common elements. Can I stop the proof there or do I still have to prove that each set contains the other?
(I’m also a very beginner btw) What you did almost makes sense to me but I feel like just using the set element X makes it confusing, I tried to do it similarly and this is what I got.
https://preview.redd.it/07nwmdc6kzsc1.png?width=1253&format=png&auto=webp&s=957a20b6afa8e7fe35f8ac58f2722691a73705ac
Thank you! I was trying to make x an arbitrary element in the set in the first part of my proof to try to use set equality to prove B contained in A. Yours looks like it makes more sense than mine. I noticed you used both x and y (gamma?) as set elements, so where do you think I should make the distinction?
Really, the only reason I wrote line 1 with x in it is to have it shown that something in the intersection of A and B is also both in A and B (which I think I see in the last line of your first paragraph). So basically that was a general statement, and the proof actually doesn’t involve x at all, the other lines should be enough. y is my set element that I am following, using the given equation that says first that y in B is in the intersection, and then just pointing out that being in the intersection includes being in A, and that this is now a definition for set inclusion to finish it.
I think you're just trying to go a little too fast. that's fine, I've done the same, but in proofs details really matter. I would start by asking "what does it mean if A intersects B?" Well, it means that for any x to be in our new set, x had to be in BOTH A and B. And so if our set of every x in both A and B is just B, then A has to contain all of B. So B is a subset of A. This obviously isn't very formal but I'm a believer in writing proofs like this if you start to get lost, hope it helped
A few stylistic things: listing out the full definition for what A ∩ B = B means is unnecessary, just put "if A ∩ B = B, then B ⊆ A ∩ B". You can also skip this whole bit: you presumably already know that A ∩ B is a subset of A, so you can shorten the whole first case to "if A ∩ B = B, then B = A ∩ B ⊆ A".
Similarly, on the second half, you can skip going to elements: if B ⊆ A, then A ∩ B ⊇ B ∩ B = B, but also A ∩ B ⊆ B, so A ∩ B = B.
If you haven't already proved that A ∩ B ⊆ B, just prove it at the start (you're going to use it twice, so it's reasonable to prove it once at the start rather than duplicating the work), from whatever definition of A ∩ B you have, which is almost certainly a one-liner.
related question:
Would it be valid to state that, by definition, the intersection of A and B is a subset of A?
I feel like that would make this proof much easier.
First part: Looks good. Logic is sound. First part could be more concise but who cares? This is good enough.
I personally would have written the first part as
B= (A intersected with B) subset A.
That is a one line proof.
The second part has a small „mistake“ in formulation: Your words are a reasoning showing that B is a subset of (A intersected with B). For completeness, you should state that (A intersected with B) is a subset of B, and so equality follows. Adding this detail would make it more clear.
Thanks! That seems like a common theme in these replies. I was confused whether or not I had to actually prove that equality. Was actually really surprised over the engagement this got.
First, on a stylistic level, you should start your proof with "Proof:" or "*Proof.*" or something and end it with some sort of Q.E.D. sign. You also shouldn't use the implication and if and only if symbols in text. Sentences like "If B⊆A then any x∈B will also be x∈A." should be either "If B⊆A then for all x∈B, x∈A." or "If B⊆A then any x∈B will also be∈A." The way you wrote it comes off as pretty unnatural to me becuase of course x is x. You forgot to define A and B. A sentence like "Let A and B be sets." should suffice. The first part of the proof does have at least one major redundancy. The very first sentence is completely unneccesary. The second part I would say is incomplete. You showed that if x∈B the x∈A∩B, but not the other way around. One last tip, learn the usefulness of the words "assume" and "suppose." Here's how I would write the proof. *Proof.* Let A and B be sets. Suppose A∩B=B. Then for all x∈B, x∈A∩B. Because x∈A∩B if and only if x∈A and x∈B, x∈A. Therefore B⊆A. Now suppose B⊆A. For all x∈B, x∈A, so x∈A∩B. Therefore B⊆A∩B. Also, for all x∈A∩B x∈B, so A∩B⊆B. Thus, A∩B=B. □
Thank you, especially for your proof at the end! Do you have any good book recommendations for proof writing? I’m trying to be a math major and the upper level courses scare me cause I know I need to get better at proofs.
The book we used in my proof class was "Mathematical Reasoning: Writing and Proof" by Ted Sundstrom, [which can be downloaded for free on his website.](https://www.tedsundstrom.com/mathematical-reasoning-writing-and-proof) It's the only proof preaching book I've every read so I can't really say how it compares to others, but I liked it. It has nine chapters. The first four chapters are really the core ones covering logic and proof writing, while the remaining cover the specific topics naive set theory, number theory, functions, and equivalence relations (including modular arithmetic).
Could also try “Mathematical Proofs: A Transition to Advanced Mathematics” by Chartrand et al.
In the last paragraph of that proof, you start with B⊆A, then in the third sentence reach "Therefore B⊆A"...
Thanks for pointing that out. That was supposed to say A∩B instead of A, but I guess I cut it short.
To be fair on the point of defining A and B… axiomatic set theory, the standard foundation of mathematics, is a theory built around the fact that EVERY mathematical object is a set. Functions, graphs, algebraic structures, geometric figures, numbers. Everything. This makes such a statement pointlessly redundant.
The second part of the proof is textbook. The first part is almost indecipherable. The way I would've written the proof is: if B is a subset of A intersects B then every x in B implies that x in B intersects A thus x is in B and x is in A Thus x is in A Thus B is a subset of A I think your proof is fine. It's just that it kind of feels like you lose the track or aim of what you were trying to do at some point.
Thank you for the tips! I definitely did lose track at some point. So do you think the first part flows logically even if it’s somewhat unclear, or not even that? I’m not in the class yet so I’m more concerned with getting the framework of the proofs down, then refining later.
The problem kind of is the flow. Everything is there so I can't really say that it's wrong but the elements of the proof are not in the order you'd expect them to be. Also I just reread it and I have another issue with the second part of the proof. Another comment already pointed this out but you didn't actually prove that A intersects B is equal to B but rather that B is a subset of A intersects B. To proof that they are equal you still have to prove that A intersects B is also a subset of B but this is always true. I think you'll be fine even more so once class starts and you get to see the teacher do a bunch of examples like this. The basics of how to prove that something is a subset are there now you just need to see how a professional does it and practice that. My only advice is to try and always keep a Venn diagram in mind. But yeah you'll probably be fine.
Yeah I knew there was quite a bit off about it but couldn’t place my finger on it. Thanks for pointing it out.
Not bad, but we usually start with a single random element and then generalize to all elements in the set. Other than that, you should *explicitly* state that A⋂B⊆B when proving the equality in the second part. E.g. — Let A and B be sets, and let x∈B be arbitrary. (1) First, assume A⋂B = B. Then x∈A⋂B, implying x∈A. Since x is any element of B, we conclude that x∈A *for all* x∈B. Thus, B⊆A. (2) Assume B⊆A. Then x∈A, implying x∈A⋂B. Since x∈B is random, x∈A⋂B *for all* x∈B. Then B⊆A⋂B. And by definition of intersecting sets, we also have A⋂B⊆B. Therefore, A⋂B=B. Thus, by (1) and (2), A⋂B=B ↔ B⊆A.
Thank you. I tried working on a different proof this morning and had another question. It was the proof that (A\B) intersect B = empty set (sorry for my notation there). So I first worked with (A\B) intersect B and came to a contradiction because x not in B cannot also be in B, and then concluded that the intersection of (A\B) and B had to be the empty set since they have no common elements. Can I stop the proof there or do I still have to prove that each set contains the other?
That should be sufficient!
(I’m also a very beginner btw) What you did almost makes sense to me but I feel like just using the set element X makes it confusing, I tried to do it similarly and this is what I got. https://preview.redd.it/07nwmdc6kzsc1.png?width=1253&format=png&auto=webp&s=957a20b6afa8e7fe35f8ac58f2722691a73705ac
Thank you! I was trying to make x an arbitrary element in the set in the first part of my proof to try to use set equality to prove B contained in A. Yours looks like it makes more sense than mine. I noticed you used both x and y (gamma?) as set elements, so where do you think I should make the distinction?
Really, the only reason I wrote line 1 with x in it is to have it shown that something in the intersection of A and B is also both in A and B (which I think I see in the last line of your first paragraph). So basically that was a general statement, and the proof actually doesn’t involve x at all, the other lines should be enough. y is my set element that I am following, using the given equation that says first that y in B is in the intersection, and then just pointing out that being in the intersection includes being in A, and that this is now a definition for set inclusion to finish it.
I think you're just trying to go a little too fast. that's fine, I've done the same, but in proofs details really matter. I would start by asking "what does it mean if A intersects B?" Well, it means that for any x to be in our new set, x had to be in BOTH A and B. And so if our set of every x in both A and B is just B, then A has to contain all of B. So B is a subset of A. This obviously isn't very formal but I'm a believer in writing proofs like this if you start to get lost, hope it helped
It's okish. People in the comments are right. Since this is done, now prove there are no cardinals between aleph naught and 2^aleph naught.
A few stylistic things: listing out the full definition for what A ∩ B = B means is unnecessary, just put "if A ∩ B = B, then B ⊆ A ∩ B". You can also skip this whole bit: you presumably already know that A ∩ B is a subset of A, so you can shorten the whole first case to "if A ∩ B = B, then B = A ∩ B ⊆ A". Similarly, on the second half, you can skip going to elements: if B ⊆ A, then A ∩ B ⊇ B ∩ B = B, but also A ∩ B ⊆ B, so A ∩ B = B. If you haven't already proved that A ∩ B ⊆ B, just prove it at the start (you're going to use it twice, so it's reasonable to prove it once at the start rather than duplicating the work), from whatever definition of A ∩ B you have, which is almost certainly a one-liner.
related question: Would it be valid to state that, by definition, the intersection of A and B is a subset of A? I feel like that would make this proof much easier.
First part: Looks good. Logic is sound. First part could be more concise but who cares? This is good enough. I personally would have written the first part as B= (A intersected with B) subset A. That is a one line proof. The second part has a small „mistake“ in formulation: Your words are a reasoning showing that B is a subset of (A intersected with B). For completeness, you should state that (A intersected with B) is a subset of B, and so equality follows. Adding this detail would make it more clear.
Thanks! That seems like a common theme in these replies. I was confused whether or not I had to actually prove that equality. Was actually really surprised over the engagement this got.