Determination of a set equality from other set equalities

[spaghetti3451]

Homework Statement

Can you conclude that A = B if we know that

(a) A \cup C = B \cup C

(b) A \cap C = B \cap C

(c) A \cup C = B \cup C and A \cap C = B \cap C

Homework Equations

The Attempt at a Solution

A=B in all three cases, but I can't find a rigorous proof for any of these cases.

Any help would be greatly appreciated.

 

[vela]

It doesn't follow that A=B in the first case. It's easy to find a counterexample. What about the other cases?

 

[spaghetti3451]

Thanks for the hint!

I've come up with these solutions based on your hint.

1. False. Counterexample: If C = ε, where ε is the universal set, then A \cup C = B \cup C implies A \cup ε = B \cup ε, which further implies that ε = ε. Therefore, A and B are allowed to be arbitrary sets and A is not necessarily equal to B.

2. False. Counterexample: If C = ø, where ø is the universal set, then A \cap C = B \cap C implies A \cap ø = B \cap ø, which further implies that ø = ø. Therefore, A and B are allowed to be arbitrary sets and A is not necessarily equal to B.

3. I'm thinking that this is true. It's always easy to find a counterexample than to find a rigorous proof for a general theorem. It would be nice if you supply a clue.

 

[vela]

As usual, show that A is a subset of B and B is a subset of A.

 

[spaghetti3451]

Thanks for the hint.

My lecturer assigned this problem as homework and I have not been able to do it without your help. What I mean that I required external guidance. Do you think that this is a sign that my intellectual capacity is not adequate to tackle these sorts of questions?

I'm sure there are students out there who could have done the problem all by themselves without requiring external guidance. I'm just trying to figure where I fall within that category of genius. :-(

 

[Tsunoyukami]

Thanks for the hint.

My lecturer assigned this problem as homework and I have not been able to do it without your help. What I mean that I required external guidance. Do you think that this is a sign that my intellectual capacity is not adequate to tackle these sorts of questions?

I'm sure there are students out there who could have done the problem all by themselves without requiring external guidance. I'm just trying to figure where I fall within that category of genius. :-(

The short answer is that I would not consider this a sign that your mental capacity is not adequate to tackle these sorts of questions.

You might be right - there are probably some students in your class who were able to solve this problem without external guidance. However, there are probably others who thought they had solved it or thought it was obvious and therefore not worth solving and they might have the wrong answer because they didn't spend any time thinking about the problem.

The process of learning requires effort on the part of the student. You made the effort to attempt the problem, ask a question to make sure you understood the problem correctly, and made another effort to solve the problem correctly and asked another question - this shows that you are engaged with the material and, at the very least, trying to understand it. THe next time you see a similar problem you might think back to this one and use what you learned as a strategy for approaching that new problem.

Some students might seem like they just magically know everything - and some of them probably do know a lot - but it's possible (and likely) that you know more than they do in another area of study.

It can be difficult when surrounded by others who are (seemingly) smarter than you but try not to compare yourself to them - instead, use them as a learning resource. It can be very helpful to have friends "smarter" than you! Good luck!

 

[vela]

My lecturer assigned this problem as homework and I have not been able to do it without your help. What I mean that I required external guidance. Do you think that this is a sign that my intellectual capacity is not adequate to tackle these sorts of questions?

Not necessarily. There are a lot of reasons why someone might struggle with a math course. For example, writing proofs presents a stumbling block for many students; if this is your first time writing proofs, it would not be surprising to me if you struggled. A sudden change in the type of math can also throw students off. Maybe, for whatever reason, you're just not at a point to learn the material, but a semester later, you will be. Perhaps the professor's teaching style doesn't match your learning style. A lack of intellectual capacity is a reason you should consider only after eliminating most other possibilities.

I'm sure there are students out there who could have done the problem all by themselves without requiring external guidance. I'm just trying to figure where I fall within that category of genius. :-(

It's a bit dangerous to compare yourself to others as their backgrounds can be quite different from yours. Some students will appear to get the material more easily simply because they've seen it before and they've had more time to practice.