Math Is Fun Forum

Kazy

Member

Registered: 2006-01-24

Posts: 37

Finding identities of binary operations

I need to determine which of the following have identities with proofs.
a) On Z (integers), n * m = 6 + nm
b) On Z, n * m = n²m²
c) On Z+, n *m = min(n,m), the smaller of n and m
d) On P(A), for any set A, X * Y = X U Y
e) On Z+, n * m =  n^m

I understand the concept of identities, i'm just unsure on how you can prove these things. I know if they don't have an identity, its easiest to show it by using a contradiction, but I'm having trouble finding any contradictions. Any help would be appreciated! Thanks.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Finding identities of binary operations

nm=6+nm?
It's false.
nm=(nm)²
so nm=0 or nm=1.
On Z+, n *m = min(n,m)
That's only if n=m and nn=n because otherwase nm>n and nm>m (n.m!=1)
so you get n=m=1
On Z+ nm=n^m
(n,1) and (2,2).

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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Finding identities of binary operations

krassi, I don't think you're trying to find the right thing.

The identity is a fixed number, e, such that:

a * e = e * a = a, for all a∈A (Where A in most of these cases is Z)

nm=6+nm

We wish to find an n such that 6+nm = m.  So solving for n, we get n = 1 - 6/m.  But n depends on m.  Thus, there exists no single n such that 6+nm = m, so no identity exists.

n * m = n²m²

Same deal here.  We want n²m² = m.  Solve for n, you get it in terms of m.

n *m = min(n,m)

Again, does not exist.  Pick any number in Z+, and you can always find a number greater than it (Archimedian Principle).  Thus, min(n, m) will never always be m for any fixed n.

X * Y = X U Y

The null set is the identity.

n * m =  n^m

No identity here because a *e = a, where e = 1, but e * a <> a.

So the majority of these have no identity.  For the set one, show that a * e = e * a = a, where a is any set and e is the null set.  It should be very striaghtforward.

However, the others are not.  You need to show that for any element you pick, there exists another element such that the element you picked can't be the identity.  I'll start working on one for you (I got some other things to do at the moment as well), but I should have it up in a few hours.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Finding identities of binary operations

Actually, nevermind.  Just solve for m like I did (for most of them), and show that this "equation" is the identity, but this equation also is not a single number.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Finding identities of binary operations

I'll do the first as an example, you try to do the rest:

1. n*m=6+nm

Proof:  Let n, m ∈ Z, such that n = 1 - 6/m.  Then:

n * m = 6 + nm = 6 + (1 - 6/m)m = 6 + m - 6 = m.
m * n = 6 + mn = 6 + m(1 - 6/m) = 6 + m - 6 = m.

Thus, n is the identity of m.  But n depends on m, and so, there is no single value n.

∴ No identity exists.

Last edited by Ricky (2006-03-08 04:12:15)

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."