# Commutative and Associative 71.5

• Oct 26th 2009, 08:41 AM
tigergirl
Commutative and Associative 71.5
Define the operation ^on Z by x^y= the smaller of x and y. For example 2^5=2, 3^-4=-4, 4^4= 4. Answer the following questions and justify your answers.

a. Is this operation commutative?
b. Is this operation associative?
c. Does this operation have an identity in Z?
• Oct 26th 2009, 08:55 AM
Chris L T521
Quote:

Originally Posted by tigergirl
Define the operation ^on Z by x^y= the smaller of x and y. For example 2^5=2, 3^-4=-4, 4^4= 4. Answer the following questions and justify your answers.

a. Is this operation commutative?
b. Is this operation associative?
c. Does this operation have an identity in Z?

The operation is commutative since $\displaystyle x \wedge y = \min\{x,y\}=\min\{y,x\}=y\wedge x$

The operation is also associative since

$\displaystyle x\wedge (y\wedge z)=x\wedge\min\{y,z\}=\min\{x,\min\{y,z\}\}=\min\{ \min\{x,y\},z\}$ $\displaystyle =\min\{x,y\}\wedge z=(x\wedge y)\wedge z$

I'm not sure what to say with regards to (c). Maybe someone else can jump in and answer that.

Does everything else make sense?
• Oct 26th 2009, 09:12 AM
TheEmptySet
To pick up where chris left off. If there were an identity $\displaystyle I$ it would need to satisfy this property

I^x=x^I=x for all $\displaystyle x \in \mathbb{Z}$

what this would imply is that I is larger than every integer.

i.e $\displaystyle I = \infty$ but infinity is not a number.

so there is not an identity.