/sci/ - Science & Math

>>5210930
Sometimes people won't include certain axioms that others do. Never heard of groups without closure, but you never know, your lecturer could have reasons in his research to want to call such a thing a group.

For example one of my professors didn't require the addition group of a ring to be abelian, while most do.

>>5210930
Bamping with Windows 7 pictures

>>5210930
>Is it always necessary that (g_1 + g_2) \in G?
Yes.

You can either have it explicitly as a group axiom or implicitly implied by saying the operation has to be well-defined.

Perhaps your teacher implied closure in some way you didn't notice, such as by saying + is a function of type <span class="math">G \times G \to G[/spoiler]?

>>5210930
The definition of a binary operation includes closure so some people dont have closure as an axiom.

>>5210941
>>5210957
>>5210960
Thank you!

>>5210960
This is not true. Simple example: the binary operator f:RxR->C given by x f y = x+iy is not closed.

what's the difference between using that symbol that looks like an epsilon to mean "is a subset of" and using the symbol that looks like a sidways U to mean "is a subset of" ?

>>5210982
A binary operator needs always be of the form <span class="math">\pi :A\times A\to A your example is just a map with two arguments.[/spoiler]

And I pretty much forgot to close the math tags lel

>>5210984
0/10

>>5210960
>>5210988
Division isn't closed over the reals. Subtraction isn't closed over the naturals.

The definition of A GROUP requires that the group's operator close (maybe indirectly defined like >>5210957 says), but a binary operation need not be.

>>5210984
the epsilon looking thing is 'an element of' and the sideways u is 'a subset of'

consider S={a,b,c,d}
a is an element of S, but not a subset.
{a} is a subset of S, but not an element of S.
{a,b} is a subset of S, but not an element of S.
the empty set is a subset, but not an element of S.
It can be a bit confusing, but hopefully that made sense

>>5210984
<span class="math">\in[/spoiler] means "is a member of", not "is a subset of"

presumably if <span class="math">x \in S[/spoiler] then <span class="math">\{ x \} \subset S[/spoiler]

>>5210994
Division is mainly multiplication by inverse on a field where clearly any <span class="math">ab^{-1}\in F[/spoiler]. Or at least, that's how the texts I've used explain.

>>5210994
Subtraction on naturals and division arent even functions on those sets. You can call them 'partial operations' maybe but they aren't technically binary operations in pretty sure unless closed.

>>5210997
Thank you. It seems arbitrary with regard to the distinction made between
>a is an element of S, but not a subset.
>{a} is a subset of S, but not an element of S.

But it does make sense.