# Can you check this proof please: sets?

• workerant
In summary, if all 4 sets are equal to the real numbers then it is not the case that x belongs to both C and D.
workerant

## Homework Statement

A,B,C,D are sets.

Prove that if C is contained in A and D is contained in B, then C∩ D is contained in A∩ B.

## The Attempt at a Solution

Let x be any element.

Then There exists (x that belongs to C∩D) and (x does not belong to A∩ B)

So x belongs to C and x belongs to D

If x belongs to C, since C is contained in A, then x belongs to A.
If x belongs to D, since D is contained in B, then x belongs to B.
So x belongs to A intersect B, a contradiction.

Then the original statement is true.

Is it okay?

workerant said:

## The Attempt at a Solution

Let x be any element.

Then There exists (x that belongs to C∩D) and (x does not belong to A∩ B)

That needn't be the case at all. Let $A=B=C=D=\mathbb{R}$.

Well, then can I say to assume that because I am trying to do a proof by contradiction.

I'm confused by what you are saying...I should say A=B=C=D=R and then what? I'm not sure I follow...is the rest okay?

Don't do proof by contradiction, it just muddles things. Everything else was fine.

So x belongs to C and x belongs to D

If x belongs to C, since C is contained in A, then x belongs to A.
If x belongs to D, since D is contained in B, then x belongs to B.
So x belongs to A intersect B,

Hence all x in C intersect D are in A intersect B, and C intersect D is a subset of A intersect B. It's much cleaner that way

workerant said:
Well, then can I say to assume that because I am trying to do a proof by contradiction.

I'm confused by what you are saying...I should say A=B=C=D=R and then what?

I was giving you a counterexample to demonstrate that your opening statement is false. If you let all 4 sets equal the real numbers then it is not the case that that there exists an $x\in C\cap D$ with $x\notin A\cap B$.

workerant said:
If x belongs to C, since C is contained in A, then x belongs to A.
If x belongs to D, since D is contained in B, then x belongs to B.
So x belongs to A intersect B, a contradiction.

I agree with Office Shredder that you should forget about proof by contradiction here, but I don't agree that this all by itself is fine. You should include a line that says that $x\in C\cap D$. After all, you're supposed to show that $(C \cap D) \subset (A \cap B)$. Those two sets should be connected by your argument.

thanks guys...I actually I did include such a line in my formal write-up so thanks

## 1. Can you explain the concept of sets and how they are used in mathematics?

Sets are a fundamental concept in mathematics that refer to a collection of objects or elements. These elements can be anything, such as numbers, letters, or even other sets. Sets are typically denoted by curly braces and the elements are separated by commas. Sets are used in various mathematical operations, such as union, intersection, and complement.

## 2. How do you prove that two sets are equal?

To prove that two sets are equal, you need to show that they have the same elements. This can be done by using the methods of set equality, such as the method of element membership or the method of subsets. You can also use mathematical notation to represent the elements of each set and show that they are identical.

## 3. Can you check my proof involving sets for any errors?

Sure, I would be happy to check your proof involving sets for any errors. Please provide me with your proof and I will carefully review it to ensure its validity and accuracy.

## 4. How can sets be used to solve real-life problems?

Sets can be used to solve real-life problems by representing the problem in terms of sets and using mathematical operations to find a solution. For example, sets can be used in probability to determine the likelihood of an event occurring, or in statistics to group and analyze data. Sets can also be used in computer science and data analysis to categorize and organize data.

## 5. What is the difference between finite and infinite sets?

A finite set is a set that has a limited number of elements, while an infinite set has an unending number of elements. For example, the set of even numbers is infinite because it has an infinite number of elements, while the set of all the days in a week is finite because it has only seven elements. Infinite sets are also divided into countable and uncountable sets, depending on whether their elements can be put into a one-to-one correspondence with the set of natural numbers.

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