JC sets question

reason vs religion

Hey. I have a question I wonder if anyone could answer for me. It's from last year's Higher JC Maths paper.

A number of identities in set notation are given and the question asks for you to say in each case whether they are: always true, sometime true (by which they ofc mean "possible") and never true.

The one I'm having trouble with is AUC = A∩C, which the marking scheme says is never true. It seems to me that if (A∩C)' is a null set, then the identity is true. Can someone set me right?

Thanks in advance

TheBody

I had a look at the actual question and for the benefit of other readers, it also says that A and C have to be different. I think this is where your problem is.

If this condition was not there, then it would be true in cases where A=C.

reason vs religion

TheBody wrote: »

I had a look at the actual question and for the benefit of other readers, it also says that A and C have to be different. I think this is where your problem is.

If this condition was not there, then it would be true in cases where A=C.

Thanks for responding and for investigating the original question! But I don't see how the condition that sets be different explains my question. They can still be different sets even if they contain the same elements or am I wrong?

TheBody

reason vs religion wrote: »

Thanks for responding and for investigating the original question! But I don't see how the condition that sets be different explains my question. They can still be different sets even if they contain the same elements or am I wrong?

We say sets are the same (or equal) if they contain the same elements. Perhaps that is your confusion. So when the questions says the sets are different, they mean that they do not contain EXACTLY the same elements.

reason vs religion

TheBody wrote: »

We say sets are the same (or equal) if they contain the same elements. Perhaps that is your confusion. So when the questions says the sets are different, they mean that they do not contain EXACTLY the same elements.

Ah, okay. Thanks for your help.

TheBody

No problem! Good luck with your studies and don't be afraid to keep the questions coming. We are a friendly bunch here on the Maths forum!!

reason vs religion

Thanks for your encouragement. But, embarrassingly, I'm actually a (non-maths) university student trying to help a sibling!

A final remark on my confusion: I see now why it makes sense to define a set by its members, helped by the below extract from university set theory notes. But, and maybe this is merely self-preservation, it seems to me the axiom of extension is more advanced than the JC material warrants. But thanks again.

The identity of a set is considered to be determined by its extension, that is, by its membership, the identity of the elements that belong to it. In all familiar variants of Axiomatic Set Theory, this assertion is called the ‘Axiom of Extension’, although it is more like a stipulation about the meaning of the term ‘set’ than a hypoth- esis of substance. With certain critical exceptions, every intension—every concept, property, and condition— determines a unique empty or non-empty set. (The postscript describing Russell’s Paradox should give a sense of the character of these exceptions). In general, arbitrarily many different intensions will determine the same set. All intensions that determine the same set are said to be co-extensive. For example, the set of all Presi- dents of the United States through the year 2005 contains the same 43 elements as does the set of all male Presidents of the United States through the year 2005—thus these two sets are identical, which means that the two defining conditions that pick out these 43 persons are co-extensive. This is true despite the fact that the two defining conditions are conceptually quite distinct, and that in another possible world differing from ours in only comparatively modest ways, these two conditions pick out two sets with distinct memberships. (A possible world is a self-consistent way things could have been. Thus there is a possible world in which life never evolved on Earth, but there is no possible world in which 2+2=5).

TheBody

Ususally in maths we try to give things sensible names or do things in the most logical way.

Here we are simply defining sets to be different if they don't contain exactly the same things (although some elements may be the same).

I don't think you should be looking for some deep meaningful axiom or theorem, just let your intution do the work. It's usually the correct thing because we try to do things in a sensible fashion