Show whether the following sets are group or not 

To determine whether a given set is a group, we must verify that it satisfies the four fundamental properties of a group under a specified operation. These properties are:

1. **Closure**: For every pair of elements \( a \) and \( b \) in the set, the operation \( * \) on \( a \) and \( b \) must yield an element that is also in the set.
2. **Associativity**: For every \( a \), \( b \), and \( c \) in the set, \( (a * b) * c = a * (b * c) \) must hold.
3. **Identity Element**: There must be an element \( e \) in the set such that for every element \( a \) in the set, \( e * a = a * e = a \).
4. **Inverse Element**: For every element \( a \) in the set, there must be an element \( b \) in the set such that \( a * b = b * a = e \), where \( e \) is the identity element.

If a set satisfies all four properties, it is a group under the given operation. If any of these properties are not satisfied, the set is not a group.

Please provide the specific set and operation you'd like to analyze, and I'll guide you through the process of determining whether it forms a group.

Explore the criteria for determining whether a given set forms a group under a specified operation. Learn about closure, associativity, identity elements, and inverse elements in group theory. Suitable for advanced undergraduate students studying abstract algebra.

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