Thread: 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

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 and
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)

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?

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.

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

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.

thank you for your help