Question

Consider a binary operation $*$ on set $\mathbb{Z}$ (set of integers) defined as $a * b = a + b + 1$ then :
A. $*$ is commutative

B. $*$ is associative

C. Identity element under $*$ exists

D. Every element has an inverse under $*$

E. The structure $(\mathbb{Z}, *)$ is not a group

Choose the correct answer from the options given below :

• A. A, B, E only
• B. B, D, E only
• C. A, C, D, E only
• D. A, B, C, D only

Hint:

To find the identity $e$ quickly, just set $a * e = a$ and solve for $e$. If $e$ is independent of $a$, it's the identity.

Answer 1

The Correct Option is D