# (a C b) and (b C a) implies a=b?

• David Carroll
In summary, the conversation discusses the concept of equal sets and the axiom of extensionality, which states that two sets are equal if and only if they are subsets of each other. The conversation also explores different ways to prove that two sets are equal, including pictorializing with Venn diagrams and using cardinality, but ultimately concludes that a formal proof based on the definition of equality is necessary.
David Carroll
Greetings. I have been studying David Poole's Linear Algebra textbook and I discovered in the Appendix that if it can be shown that if the set A is a sub-set of B and B is a sub-set of A, then the set A is exactly the same as the set B. And this all seems intuitively plausible, but for the life of me I couldn't prove it. No proof was adduced in the textbook, but I assume such a proof exists.

Is it simply an axiom or is it derived from something else?

I tried to take it along these lines: Some of the elements of B form the entirety of the elements of A. And the some of the elements of B form the entirety of the elements of B.

I even tried to pictorialize it with an ad hoc variation of the Venn diagram, where the bubble representing B curled around and entered A in a sort of two-dimensional Klein bottle. No help here either.

Any suggestions?

Try to assume the opposite and show that it leads to a contradiction.

Hmmmm. I'll try.

Okay, let Na represent the number of elements of the set A. Let Nb represent the number of elements of set B.

Then, if every element of set A is an element of set B it follows that Na is less than or equal to Nb. Also, if every element of set B is an element of set A, it follows that Nb is less than or equal to Na.

But if Na < Nb then Nb cannot < Na, therefore the number of elements of B equals the number of elements of A.

Does that work? Or is something missing?

David Carroll said:
Greetings. I have been studying David Poole's Linear Algebra textbook and I discovered in the Appendix that if it can be shown that if the set A is a sub-set of B and B is a sub-set of A, then the set A is exactly the same as the set B. And this all seems intuitively plausible, but for the life of me I couldn't prove it. No proof was adduced in the textbook, but I assume such a proof exists.

Is it simply an axiom or is it derived from something else?

I tried to take it along these lines: Some of the elements of B form the entirety of the elements of A. And the some of the elements of B form the entirety of the elements of B.

I even tried to pictorialize it with an ad hoc variation of the Venn diagram, where the bubble representing B curled around and entered A in a sort of two-dimensional Klein bottle. No help here either.

Any suggestions?

I'm not sure what you are trying to picture. A = B clearly meets the criteria. Remember that "subset of" means "proper subset of or equal to".

There can be no other option. B cannot be a proper subset of A, because then A cannot be a subset of B, so B must be equal to A.

You can in fact take this as the definition of equality for sets:

##A = B## iff ##A \subset B## and ##B \subset A##

It's a similar argument to: ##a \le b## and ##b \le a## implies ##a = b##

David Carroll said:
Hmmmm. I'll try.

Okay, let Na represent the number of elements of the set A. Let Nb represent the number of elements of set B.

Then, if every element of set A is an element of set B it follows that Na is less than or equal to Nb. Also, if every element of set B is an element of set A, it follows that Nb is less than or equal to Na.

But if Na < Nb then Nb cannot < Na, therefore the number of elements of B equals the number of elements of A.

Does that work? Or is something missing?
No. Two different sets can have the same number of elements. So proving that the sets have the same cardinality will not prove that the sets are equal. For a formal proof, you need to rely on the formal definition of equality of 2 sets. I am sure you can prove it directly or by contradiction. Assume A and B are different. Does that mean there is an element in one that is not in the other? Then where does that lead you?

David Carroll
That last one did it for me. Thank you, FactChecker.

PeroK said:
I'm not sure what you are trying to picture. A = B clearly meets the criteria. Remember that "subset of" means "proper subset of or equal to".

There can be no other option. B cannot be a proper subset of A, because then A cannot be a subset of B, so B must be equal to A.

You can in fact take this as the definition of equality for sets:

##A = B## iff ##A \subset B## and ##B \subset A##

It's a similar argument to: ##a \le b## and ##b \le a## implies ##a = b##
Yup, that's right. But this is wrong ##a < b, b < a \Rightarrow a = b##

## 1. How is the statement "(a C b) and (b C a) implies a=b" related to mathematics?

This statement is related to the transitive property in mathematics, which states that if a is related to b and b is related to c, then a is also related to c. In this case, the relation is equality, so if a is equal to b and b is equal to a, then a must also be equal to b.

## 2. Why is the statement "(a C b) and (b C a) implies a=b" important in mathematics?

This statement is important because it helps us understand the fundamental concept of equality. It allows us to make logical conclusions about the relationships between different quantities based on their equality.

## 3. Can you give an example to illustrate the statement "(a C b) and (b C a) implies a=b"?

Sure, for example, if we have the statements "2 is less than 5" and "5 is less than 8", we can use the transitive property to conclude that "2 is less than 8". In this case, the relation is less than, but the same concept applies to equality.

## 4. Is the statement "(a C b) and (b C a) implies a=b" always true?

Yes, this statement is always true. The transitive property is a fundamental property of equality, and it holds true for all values of a and b.

## 5. How can we use the statement "(a C b) and (b C a) implies a=b" in real-life situations?

In real-life situations, we can use this statement to make logical conclusions about equality. For example, if we know that two sides of a triangle are equal and the third side is also equal to one of the other sides, we can use this statement to conclude that all three sides are equal.

Replies
2
Views
1K
Replies
4
Views
2K
Replies
10
Views
2K
Replies
9
Views
4K
Replies
1
Views
2K
Replies
2
Views
904
Replies
2
Views
2K
Replies
7
Views
3K
Replies
3
Views
969
Replies
2
Views
3K