Thread: A set in a set?

Hi!

I've just taken on a course in discrete mathematics, i think im doing fairly well but there is one thing i really don't understand about sets and i thought maybe someone here could help me out with it.

A= {a,b,c}
B= {a,b,g}
C= {c,d,e,f}
G= {a,b,c,d,e,f,g}

{a,b,{g}} = B true/false?

What does this last line mean? (The left side of the =)

I can't find it in my litterature either. :/

Thanks in advance!

(Well, sets may contain other sets, that's not very upsetting - you 'll be used to it in no time)


When writing {a,b,{g}}, you have formed a set with elements a,b and {g} - careful, not just g, but the one-element set {g} (called a "singleton set").

So, writing "a belongs to {a,b,{g}}" is right, but "g belongs to {a,b,{g}}" is wrong; Actually {g} belongs to {a,b,{g}}.

Try now to show that {a,b,{g}} is not equal to B, keeping the singleton in mind!

Uhhm, ok. Here goes nothing.

Since {g} is not equal to g, {a,b,g} is not equal to {a,b,{g}} and therefore, {a,b,{g}} is not equal to B?

Equality of sets means they share the same elements,
and as you said is false. The sets are not equal.

Another method to use is given two sets , if you can show then .
Thus, show that each set is a subset of each other. Which is not true here.

B= {a,b,g}


{{b}} belongs to B
{{b}} is a subset of B
{{b}} is a subset of the powerset of B

I hope i understood everything correctly.
According to "what i know" i would say that all of the above statements are false, is that correct?
If it would be {b} instead of {{b}}, they would all be true?

B= {a,b,g}


{{b}} belongs to B
{{b}} is a subset of B
{{b}} is a subset of the powerset of B

For this set B,

- b belongs to B (drop curly brackets once to obtain elements)

- {b} is a subset of B (enclosing elements in brackets, forms a subset)

- {{b}} is a subset of the powerset
P(B)={ {a},{b},{g},{a,b},{a,g}{b,g}{a,b,g} }

and, what is the relation of {b} tp P(B)?

{b} is a proper subset of P(B) as far as i know.

So {{b}} is a subset of P(B) because the element of {{b}}, {b} that is, is included in P(B) and therefore, {{b}} is a subset of P(B)?

Just checking, P(B) stands for "the powerset of B" right?

Hi Rebesques:

I suspect this was just an oversight (you obviously know your stuff) but,
nP(S) = 2^(nP(S)) for any set, S, where n denotes "cardinality of". Thus
nP(B) = 2^nB = 2^3 = 8. You omitted {} from the power set.

Regards,

Rich B.

{b} is a proper subset of P(B) as far as i know.

So {{b}} is a subset of P(B) because the element of {{b}}, {b} that is, is included in P(B) and therefore, {{b}} is a subset of P(B)?

Just checking, P(B) stands for "the powerset of B" right?

Everything is ok, except from the first sentence. it is {{b}} that is a proper (containing less!) subset of P(B).

=========================================

You omitted {} from the power set.

Yeah you are right Rich. I forget stuff all the time...
(better it be the empty set than my wallet )

In relation to all this, just WHY can we include {g} in a set along with the elements a and b? I realize we CAN do this, but doesn't it create some sort of logical equivalence of the "nature" (for lack of a better word) of the element a to the nature of the set {g}? To say an element and a set have some feature in common bothers me. It would seem to me that all the elements of a set would need to have the same "nature." (And that way we don't run into things like Russell's paradox...we couldn't form the set of all sets and call it a set...it would need to be called something else.)

-Dan