1. **Stating the problem:** Given the operation $a * b = a + b - 1$, we want to understand how this operation works and possibly find some properties or results using it. 2. **Understanding the operation:** The operation $*$ takes two numbers $a$ and $b$ and combines them by adding them together and then subtracting 1. 3. **Example calculation:** For example, if $a=3$ and $b=4$, then $$3 * 4 = 3 + 4 - 1 = 6.$$ 4. **Checking properties:** Let's check if this operation is commutative, i.e., if $a * b = b * a$. $$a * b = a + b - 1$$ $$b * a = b + a - 1$$ Since addition is commutative, $a + b = b + a$, so $a * b = b * a$. 5. **Associativity check:** Check if $(a * b) * c = a * (b * c)$. Calculate left side: $$(a * b) * c = (a + b - 1) * c = (a + b - 1) + c - 1 = a + b + c - 2$$ Calculate right side: $$a * (b * c) = a * (b + c - 1) = a + (b + c - 1) - 1 = a + b + c - 2$$ Both sides are equal, so the operation is associative. 6. **Identity element:** Find $e$ such that $a * e = a$. $$a * e = a + e - 1 = a$$ Solve for $e$: $$e = 1$$ So, the identity element is 1. 7. **Inverse element:** Find $b$ such that $a * b = e = 1$. $$a + b - 1 = 1$$ $$b = 2 - a$$ **Final summary:** The operation $a * b = a + b - 1$ is commutative and associative with identity element 1 and inverse of $a$ is $2 - a$.