Here are some exercises. Fill in the blanks with complete sentences.

(I) For any integers a,b, define a + b = a-b. Prove is an operation on Z.

Solution. Choose a, b e Z. Note that a+b=a-b, which is also an integer.

(II) For any positive integers a, b, define ab=a-b. Prove is not an operation on Zo.

Solution.

Therefore, we see that is not an operation on Zo

(III) For any a, b e Qo, define a ob= a/b, Then o is an operation on Q.

Solution. Choose any a, b € Q.o.

We see that

000-20-4-20

a ob= O

=

d

fdf

ce

0

Since c, d, e, f are nonzero as we noted above, we see that cd, df are nonzero, so that Q₂0.

Therefore, we conclude that is an operation on Q0.

(IV) Consider for a moment the operation of multiplication on the integers. Prove that this

operation admits an identity.

Thus, we see that 1 is the identity element for multiplication

Solution.

on the integers.

Finally, here is an opportunity to write a complete proof: