Solution

The correct answer is ∗(a, b) = a  + b + 2

Explanation:

1) Define: * (a, b) = a + b + 2

• Closure: If you add any two integers, a and b, then add 2, you still get an integer. So, this operation is closed.
• Associativity: Check if * (a, * (b, c)) is the same as * (* (a, b), c):
  • First, find * (b, c): b + c + 2
  • Now, * (a, (b + c + 2)): a + (b + c + 2) + 2 = a + b + c + 4
  • Next, find * (a, b): a + b + 2
  • Now, * ((a + b + 2), c): (a + b + 2) + c + 2 = a + b + c + 4
  • Since a + b + c + 4 is equal to a + b + c + 4, the operation is associative.
• Identity: Find an element e such that * (a, e) = a for all a:
  • a + e + 2 = a
  • Solving for e: e + 2 = 0 which gives e = -2
  • The identity element is -2.
• Invertibility: For each a, find b such that * (a, b) = -2:
  • a + b + 2 = -2
  • Solving for b: b = -4 - a
  • For each a, b = -4 - a is an integer, so invertibility holds.

This operation forms a group.

2) Define: * (a, b) = 2a + b

• Closure: If you multiply an integer a by 2 and then add another integer b, you get an integer. So, this operation is closed.
• Associativity: Check if * (a, * (b, c)) is the same as * (* (a, b), c):
  • First, find * (b, c): 2b + c
  • Now, * (a, (2b + c)): 2a + (2b + c) = 2a + 2b + c
  • Next, find * (a, b): 2a + b
  • Now, * ((2a + b), c): 2(2a + b) + c = 4a + 2b + c
  • Since 2a + 2b + c is not equal to 4a + 2b + c, the operation is not associative.

This operation does not form a group.

3) Define: * (a, b) = (a + b) / 2

• Closure: If a and b are integers, (a + b) / 2 is not always an integer (for example, if a = 1 and b = 2). So, this operation is not closed.

This operation does not form a group.

4) Define: * (a, b) = a² + b

• Closure: If a and b are integers, a² + b is an integer. So, this operation is closed.
• Associativity: Check if * (a, * (b, c)) is the same as * (* (a, b), c):
  • First, find * (b, c): b² + c
  • Now, * (a, (b² + c)): a² + (b² + c) = a² + b² + c
  • Next, find * (a, b): a² + b
  • Now, * ((a² + b), c): (a² + b)² + c
  • Since a² + b² + c is not equal to (a² + b)² + c, the operation is not associative.

This operation does not form a group.
So, Out of the given operations, only * (a, b) = a + b + 2 forms a group on the set of integers Z.