# Proving Sets

• Feb 26th 2010, 08:04 PM
murielx
Proving Sets
A and B are sets i have to show proof that the following are equivalent

(i) A(is a subset of ) B
(ii) A(intersect)B = A
(iii) A(union) B = B

*note with (i) the symbol used is the subset symbol with a line underneath and im not sure if that means A=B or not.

Thanks
• Feb 26th 2010, 08:18 PM
Bacterius
Note that as (i) implies that A is a subset of B, then it follows that the intersection of B with A equals A, and it also follows that the union of A with B is B. The other way round : if the union of A with B is B, then A must be a subset of B, and it follows that the intersection of B with A equals A.

To help understand this, draw a diagram with $A = \{4, 5, 7\}$ and $B = \{2, 3, 4, 5, 6, 7, 8, 9\}$ :)
Once you will have understood why this works, you will be able to put it in mathematical terms.

Here are some basic ideas that can help you, too :

If A is a subset of B, it means that all the elements of A are contained in B, among other elements of B
If A intersect B = A, it means that the only elements that are in A and in B are the elements of B (equivalent to : A is a subset of B)
If A union B = B, it means that the elements that are in A or in B, are equal to the elements of B (equivalent to : A is a subset of B)
• Feb 27th 2010, 09:01 PM
murielx
thank you.
I understand the relationships between the different sets, however I am unsure as to how it should be represented as a proof.
Using what you said I can write:

(i) implies that A is a subset of B and if x (is in) A then x (is in) B

A(intersection) B = A implies that
if x (is in) A and x (is in) B then x (is in) B = A(is a subset of) B

A(union)B = B implies that
if x (is in) A or x (is in) B then x (is in) B = A(is a subset of) B

therefore (i), (ii), (iii) are equivalent

is this a proof or not?
• Feb 27th 2010, 09:22 PM
Bacterius
Sketched proof :

(1) If A is a subset of B, then A intersect B = A
(2) If A is a subset of B, then A union B = B
(3) Therefore, by transitivity, if A is a subset of B then A intersect B = A and A union B = B.

You only need to prove the weaker statements (1) and (2). Here is an example for (1) :

Statement : If A is a subset of B, then A intersect B = A

Proof : A is a subset of B, therefore each element of A is in B. This also implies that there exists no element of A which is not in B, and it follows that if A is a subset of B, then A intersect B = A.

(2) can be proven in a similar way, and once you will have proven it you will have proven your original problem.
• Feb 28th 2010, 12:30 AM
murielx
how is this for proving (2):

A is a subset of B

therefore if x is in A then x is in B

this also implies that if x is in A or x is in B then x is in B.

and it follows that if A is a subset of B then A union B = B

the proof sounds a bit off to me
• Feb 28th 2010, 12:52 AM
Bacterius
Yes, it is correct. Check out this one.

Statement : If A is a subset of B, then A union B = B

Proof : A union B returns the set of elements that are in A or in B. Say that A is a subset of B. It follows that any element that is in A is in B, and that the elements of B that are not in A are only in B. It immediately follows that A union B = B.

Of course you can do it using letters, it equally works.
• Feb 28th 2010, 01:22 AM
murielx
thank you for your help