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

Concept:
• A binary operation \(*\) on a set \(G\) forms a group if it satisfies four properties: Closure, Associativity, Identity, and Inverse.
• If it also satisfies Commutativity, it is an Abelian group.

Step 1: Verify Commutativity (A)
We check if \(a * b = b * a\) for all \(a, b \in Z\).
\(a * b = a + b + 1\)
\(b * a = b + a + 1\)
Since integer addition is commutative (\(a + b = b + a\)), \(a * b = b * a\) is true. Statement A is correct.

Step 2: Verify Associativity (B)
We check if \((a * b) * c = a * (b * c)\).
LHS: \((a * b) * c = (a + b + 1) * c = (a + b + 1) + c + 1 = a + b + c + 2\).
RHS: \(a * (b * c) = a * (b + c + 1) = a + (b + c + 1) + 1 = a + b + c + 2\).
LHS = RHS. Statement B is correct.

Step 3: Find the Identity Element (C)
Let \(e\) be the identity such that \(a * e = a\).
\(a + e + 1 = a \implies e + 1 = 0 \implies e = -1\).
Since \(-1\) is an integer (\(-1 \in Z\)), the identity element exists. Statement C is correct.

Step 4: Find the Inverse for every element (D)
Let \(b\) be the inverse of \(a\) such that \(a * b = e\).
\(a + b + 1 = -1 \implies b = -a - 2\).
For any integer \(a\), the value \(-a - 2\) is also an integer. Statement D is correct.

Step 5: Determine if it is a group (E)
Since properties A, B, C, and D are satisfied, \((Z, *)\) is indeed a group.
Therefore, statement E ("is not a group") is false.