The image you've uploaded contains a table related to algebraic structures. The task is to determine whether the operations defined in the table satisfy the properties of different algebraic structures, specifically groupoids, semigroups, monoids, and groups.

Here’s a breakdown of what each algebraic structure requires:

1. Groupoid: For each pair of elements, the operation must be closed (the result of the operation must still belong to the set).
2. Semigroup: The operation must be closed and associative.
3. Monoid: It must satisfy the semigroup conditions and also have an identity element.
4. Group: It must satisfy all the conditions of a monoid and, in addition, every element must have an inverse.

The table is organized as follows:

• Column 1 lists the sets being considered (e.g., $\mathbb{Z}$, $\mathbb{N}$, etc.).
• Column 2 defines the operation being considered (e.g., addition, subtraction, etc.).
• The next columns are to determine whether each operation satisfies the requirements for groupoids, semigroups, monoids, and groups, respectively.

Let me know if you'd like me to walk through solving this, or if you would like to focus on any specific section of the table.

Would you like a detailed step-by-step breakdown for one of the sets or operations, or should I analyze the entire table?

Tip: When analyzing algebraic structures, always start by checking closure and associativity. These are key properties for understanding the structure.

Here are 5 relevant questions:

1. Do you need a clarification on what each algebraic structure represents?
2. Would you like examples of how these operations can be tested for associativity?
3. Should I provide details on how to check for identity elements in monoids?
4. Would you like to know how to verify if inverses exist for groups?
5. Do you need any help with the operations involving matrix sets?