Return to Homework 2, Glossary, Theorems

### Problem 8

Determine if the definition of $*$ does give a binary operation on the set. If it does not, state whether condition 1, condition 2, or both conditions given in class are violated.

(a) On $\mathbb{Z}^+$, define $*$ by $a*b=a-b$

(b) On $\mathbb{Z}^+$, define $*$ by $a*b=c$, where $c$ is the largest integer less than the product of $a$ and $b$.

**Solution**

(a) This is not a binary operation. Let $a, b \in \mathbb{Z}^+$. Suppose $b>a$. This will produce a negative integer, which clearly is not in $\mathbb{Z}^+$. Therefore $\mathbb{Z}^+$ is not closed under $*$, violating condition 2. $\blacksquare$

(b) It is not a binary operation. It violates condition 2, $1*1 = 0, 0\notin\mathbb{Z}^+$.