Help with sets

• Jul 23rd 2013, 11:25 PM
mandarep
Help with sets
Hi guys,
Hopefully this is the right section, but I've got a question to with sets.

I've been given 4 separate sets that differentiate only a little and I was hoping someone could let me know how they are different?

A={a, b}
B={{a}, {b}}
C= {a, b, {a, b}}
D= {{a}. {b}. {b, a}}

To me they all look really similar but I just don't know how to explain the differences. So if someone could help me that would be awesome!
Thanks
• Jul 24th 2013, 01:04 AM
Gusbob
Re: Help with sets
Quote:

Originally Posted by mandarep
Hi guys,
Hopefully this is the right section, but I've got a question to with sets.

I've been given 4 separate sets that differentiate only a little and I was hoping someone could let me know how they are different?

A={a, b}
B={{a}, {b}}
C= {a, b, {a, b}}
D= {{a}. {b}. {b, a}}

To me they all look really similar but I just don't know how to explain the differences. So if someone could help me that would be awesome!
Thanks

I'll explain the difference between A and B. Hopefully you can work out the rest by yourself.

First, we see that $A=\{a,b\}$. What does this mean in human language? This says that the set A has two elements: a and b.

Next, we have that $B=\{\{a\},\{b\}\}$. This says that the set B has two elements, {a} and {b}.

I suppose your confusion then comes from the difference between a and {a}. It may be useful to think of a set as a plastic bag and the elements a,b as coins (perhaps a 10cent and 5 cent coin). Using this analogy, $A=\{a,b\}$ is a plastic bag containing the a 5 cent piece and a 10 cent piece. That is, when you open the plastic bag A, you find the two coins a and b. On the other hand $B=\{\{a\},\{b\}\}$ is a plastic bag containing two other plastic bags (each containing one type of coin).

In view of this analogy, you can say that A and B are actually different sets: one contains the elements a,b, the other contains the singletons of a,b (we call single element sets singletons).

Of course, you must be aware that this analogy is not perfect: The set {a,a} and {a} are equivalent since a set cannot have repeat elements. So everything in your plastic bag must have some distinguishing feature to tell them apart. For example {a,{a}} is a two element set since a is a 10 cent coin and {a} is a plastic bag wrapped around a 10 cent coin, so you can distinguish between the two).

I hope this helps.
• Jul 24th 2013, 01:22 AM
mandarep
Re: Help with sets
Thank you so much for that explanation! That analogy actually does make it much clearer.
So ultimately, none of the four sets above are the same correct?
• Jul 24th 2013, 04:06 AM
Gusbob
Re: Help with sets
Quote:

Originally Posted by mandarep
Thank you so much for that explanation! That analogy actually does make it much clearer.
So ultimately, none of the four sets above are the same correct?

Yes
• Jul 24th 2013, 05:49 AM
HallsofIvy
Re: Help with sets
In D= {{a}. {b}. {b, a}}, what does the dot, rather than a comma, mean? I don't recognize that notation.