Math Is Fun Forum

Seetha Rama Raju Sanapala

Member

Registered: 2016-10-23

Posts: 4

group with union or intersection as the operation

I want to know if we can have a group of at least 2 elements with sets as elements and some set operation (union, intersection or some other) as the binary operation.

Lehona

Member

Registered: 2016-10-22

Posts: 16

Re: group with union or intersection as the operation

I could only think of the complement, but that doesn't fulfill the associativity requirement.
You could argue that in modular arithmetic, every element (e.g. of (Z_7, +)) is actually a set, but the operation wouldn't be one of the usual set operations.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: group with union or intersection as the operation

hi Seetha Rama Raju Sanapala

Welcome to the forum.

I am replacing an earlier post with this version.  My apologies to anyone who spent time on my previous version.  Hopefully, I've got it right now.

Note:  both intersection and union are associative. 

I'll consider intersection first.

If intersection is the binary operation for a group, there must be an identity set;  let's call it I.

Also we want another member; let's call it A.

For a group we require inverses; let's call the inverse of A, B.  So

This means that I is contained in A {and also in B).

But I is the identity so

So A is contained in I.

So A is contained in I and I is contained in A.  This means that A = I.

So every element of the group is I.  Conclusion: no such groups with 2 or more members.

The case for union follows similar lines.

So A is contained in I

But I is an identity so

So I is contained in A.  etc etc.

Bob

ps.  Sorry but your full name is too long to fit the blue space above.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Seetha Rama Raju Sanapala

Member

Registered: 2016-10-23

Posts: 4

Re: group with union or intersection as the operation

Dear Bob Bundy,

Thanks a lot for the clear proof of impossibility of union and intersection as the operation of a group.  But there can be other set operations like difference, symmetric difference, complement of sets etc -  any thing based on the elements of the sets .  Can we show that they are all ruled out?

Thanks,
Seetha Rama Raju Sanapala

Lehona

Member

Registered: 2016-10-22

Posts: 16

Re: group with union or intersection as the operation

Symmetric Difference should be doable, right? A o B = B o A, ∅ is the neutral element and A^(-1) = A.
Closure should be achievable, too, right? Let's say A o B = C, then A o C = B -> Whatever you do, you still land in the group. It's late at night here, so please someone correct me if I'm wrong, but it sounds plausbile in my head.