#### jonboy

##### Full Member
Hello FreeMathHelp! What a wonderful day it is! Now if you can help me understand this problem:

Jackie claims that there are identity properties for addition, subtraction, and multiplication. Which of the following statements is correct?

.....A. Her assertion is correct because all three operations have an identity property.
.....B. Her assertion is false because subtraction does not have an identity property.
.....C. Her assertion is false because multiplication does not have an identity property.
.....D. Her assertion is false because addition does not have an identity property.

I choose A because:

$$\displaystyle \L \;9\,+\,0\,=\,9$$

$$\displaystyle \L \;9\,-\,0\,=\,9$$

$$\displaystyle \L \;9\,\bullet\,1\,=\,9$$

The answer is B. Why is this?

#### tkhunny

##### Moderator
Staff member

Dr. Math includes it, but doesn't quite say it.

http://mathforum.org/dr.math/faq/faq.pr ... ssary.html

A recent Algebra 1 text makes no mention of requiring commutativity. Bellman, Bragg, et. al. Prentice Hall (2004)

Group Theory requires commutativity for an Identity, but not for the entire group, unless it is Abelian

http://members.tripod.com/~dogschool/groups.html

Herstein, Topics in Algebra, Blaisdell (1964) agrees exactly with the just previous description.

So, identities are commutative, even in nonabelian groups.

#### pka

##### Elite Member
Jonboy,
There is a very simple reason why B is the correct answer.
Technically there are only two operations in a field, the real numbers form a field.
The operations are addition and multiplication. So subtraction is not an operation.
We define identity elements only for the two operations on the set.