A set and subset. Can anyone explain what the difference?

[ovoleg]

Homework Statement

If A is a subs4et of B and B is a subset of C, then A is a subset of C. But if A = {3}, B={{3},5}, and C = {B, 17}, then A is contained in B and B is contained in C, but A is not contained in C. The set C has exactly two members, and it is easy to see that neither of these members is a set of A

Homework Equations

Why? if B = {{3}, 5} then shouldn't C be = {{3}, 5, 17} ??

The Attempt at a Solution

I don't see how A is not contained in C

Can someone clarify?

Thanks!

 

[electron]

well..i think A is a subset of C...donno.why r u saying its not true.

 

[ovoleg]

well..i think A is a subset of C...donno.why r u saying its not true.

A IS a subset of C. That is not what I am stating,

I am saying A is not contained in C.

Hold on let me try to find the LaTeX symbol thingie...

 

[HallsofIvy]

Homework Statement

If A is a subset of B and B is a subset of C, then A is a subset of C. But if A = {3}, B={{3},5}, and C = {B, 17}, then A is contained in B and B is contained in C, but A is not contained in C. The set C has exactly two members, and it is easy to see that neither of these members is a set of A

Okay, in the second case, A is a member of B, not a subset. Also, in that case, B is a member of C, not a subset.

Homework Equations

Why? if B = {{3}, 5} then shouldn't C be = {{3}, 5, 17} ??

Well, this is an example. C can be anything that demonstrates the point! In the given example, B is a member of C, not a subset. They you want it, C= {{3},5,17}, B is a subset of C, not a member. Of course, even if B is a subset of C, since A is not a subset of B, it would not follow that A must be a subset of C, so either way makes the point.

The Attempt at a Solution

I don't see how A is not contained in C

Can someone clarify?

Thanks!

What do YOU mean by "contained in"? Subset or member? "A member of" and "a subset of" are completely different concepts. You should use one of those and not "contained in" which is ambiguous. In the example as given, C= {B, 17}, B is a member of C while A is neither a member of C nor a subset of it. In your example, C= {{3},5,17}, B is a subset of C and A is a member of C but not a subset of C.

Notice, by the way, that in the "theorem" you state: "If A is a subset of B and B is a subset of C, then A is a subset of C" if either of the hypotheses is not true (A is not a subset of B or B is not a subset of C) then the conclusion "A is a subset of C" is not necessarily true. But it still might happen to be true! A really strange example would be with A= {3}, B= {{3},5}, C= {{3},5, 3}. The A is a member of B, not a subset. B is a subset of C and A is both a member of C (because of the {3} in C) and a subset (because of the 3 in C).

 

[ovoleg]

REVISED:

If A \subseteq B and B \subseteq C, then A \subseteq C. But if A = {3}, B={{3},5}, and C = {B, 17}, then A \in B and B \in C, but A NOT \in C. The set C has exactly two members, and it is easy to see that neither of these members is a set of A

 

[ovoleg]

What do YOU mean by "contained in"? Subset or member? "A member of" and "a subset of" are completely different concepts. You should use one of those and not "contained in" which is ambiguous. In the example as given, C= {B, 17}, B is a member of C while A is neither a member of C nor a subset of it. In your example, C= {{3},5,17}, B is a subset of C and A is a member of C but not a subset of C.

Notice, by the way, that in the "theorem" you state: "If A is a subset of B and B is a subset of C, then A is a subset of C" if either of the hypotheses is not true (A is not a subset of B or B is not a subset of C) then the conclusion "A is a subset of C" is not necessarily true. But it still might happen to be true! A really strange example would be with A= {3}, B= {{3},5}, C= {{3},5, 3}. The A is a member of B, not a subset. B is a subset of C and A is both a member of C (because of the {3} in C) and a subset (because of the 3 in C).

I see what you are saying, but let me see if I can understand this logic.

If B is a subset of C then all members of B are also members of C. If all members that are in A are contained in B then wouldn't that imply that all the members in A are part of C. So saying A \in C is not correct because? You would then have to have C= {A, 17} for A \in C to be correct?? So the set has to be a member? for that notation to be correct?