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

We check the group axioms for $(\mathbb{Z}, *)$. 1. Commutativity (A): $a * b = a + b + 1$ and $b * a = b + a + 1$. Since addition is commutative, $a * b = b * a$. Correct. 2. Associativity (B): $(a * b) * c = (a + b + 1) * c = (a + b + 1) + c + 1 = a + b + c + 2$. $a * (b * c) = a * (b + c + 1) = a + (b + c + 1) + 1 = a + b + c + 2$. LHS = RHS. Correct. 3. Identity Element (C): $a * e = a \implies a + e + 1 = a \implies e = -1$. Since $-1 \in \mathbb{Z}$, identity exists. Correct. 4. Inverse Element (D): $a * a' = e \implies a + a' + 1 = -1 \implies a' = -a - 2$. Since $-a-2$ is an integer for any $a$, inverse exists. Correct. Conclusion: The structure $(\mathbb{Z}, *)$ satisfies all group axioms. Thus, E is incorrect. A, B, C, D are correct.