Question

Maybe stupid question, but why is \(\{n^2~|~n \in \mathbb{Z}\}=\{{0,1,4,9,16,25,\cdots\}}\), not \(\{0,1,1,4,4,9,9,16,16,25,25,\cdots\}\)?

Answer 1

May I know why do you think it should be {0,1,1,4,4...} ?

Answer 2

Because there is both negative and positive integer to square, so there should be duplicate elements except 0

Answer 3

given that \(\mathbb{Z}=\{0,1,-1,2,-2,3,-3,\cdots\}\)

Answer 4

Ahh I see...

Answer 5

guess i would google and see if repetitions are allowed in set notation

Answer 6

ah I see, according to this link: http://cr.yp.to/2005-261/bender1/SF.pdf

"
{a, b, c} = {b, a, c} = {c, b, a} = {a, b, b, c, b}.

The first three are simply listing the elements in a different order. The last happens to
mention some elements more than once. But, since a set consists of distinct objects, the
elements of the set are still just a, b, c. Another way to think of this is:

Two sets A and B are equal if and only if every element of A is an
element of B and every element of B is an element of A."

Answer 7

So {0,1,1,4,4,9,9,16,16,25,25,⋯} is correct, but can be reduced to {0,1,4,9,16,25,⋯}

It makes sense, I think

Answer 8

right? @SithsAndGiggles @rational

Answer 9

Yeah duplicate elements are typically consolidated. The set \(\{1,1\}\) is the same as \(\{1\}\).

Answer 10

I see. Interesting...

Thanks for clarification!

Answer 11

its the same group :P