# Thread: Set Equality and Cardinality

1. ## Set Equality and Cardinality

Let me define two sets:
A={1}
B={1,1}

Few questions:
1. Is A=B?
2. How many elements (or cardinality) does A have? What about B?
3. Is there any time in mathematics that we need to make a distinction between A and B?

(I guess I am just confused, so any guidance would be welcome)
Thanks,

2. Originally Posted by aman_cc
Let me define two sets:
A={1}
B={1,1}

Few questions:
1. Is A=B?

$\color{red}\mbox{Yes. In sets we have no multiplicity, so you can put 1 million 1's, it's the same as A}$

2. How many elements (or cardinality) does A have? What about B?

$\color{red}\mbox{as } A=B \,\,\,\,\mbox{then...}$

3. Is there any time in mathematics that we need to make a distinction between A and B?

$\color{red}\mbox{As sets and in the usual frame of set theory axioms the answer is no}$

$\color{red} Tonio$

(I guess I am just confused, so any guidance would be welcome)
Thanks,
.

3. A set is complelety defined by its elements, and since same name means same object (I guess it is quite reasonnable in mathematics): in $\{1,1\},$ there is only one thing, which is $1,$ thus $\{1,1\}=\{1\}$.

When you want to consider a collection of objects and be able to see them appear multiple times, you can use families or sequences or tuples.

EDIT: Ah Tonio was faster

4. Originally Posted by clic-clac
A set is complelety defined by its elements, and since same name means same object (I guess it is quite reasonnable in mathematics): in $\{1,1\},$ there is only one thing, which is $1,$ thus $\{1,1\}=\{1\}$.

When you want to consider a collection of objects and be able to see them appear multiple times, you can use families or sequences or tuples.

EDIT: Ah Tonio was faster
Thanks clic-clac and Tonio. Really appreciate your help.

5. 3. Is there any time in mathematics that we need to make a distinction between A and B?
Yes, in multisets: Multiset - Wikipedia, the free encyclopedia, but I have never really worked explicitly with them.

6. Originally Posted by aman_cc
3. Is there any time in mathematics that we need to make a distinction between A and B?
There is also something related called the disjoint union. It indexes elements based on which sets they came from. For instance, if $A_1=A_2=\{1\}$,

$A_1\cup A_2=\{1\}$ (regular union)

$A_1\sqcup A_2=\{(1,1),(1,2)\}$ (disjoint union)

(Note that those should be different union symbols, though they look sort of similar.)