Given the mathematical system defined in the table below, determine if the system is a group.

StartLayout 1st Row 1st Column asterisk 2nd Column 0 3rd Column 2 4th Column 4 2nd Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 3rd Row 1st Column 2 2nd Column 0 3rd Column 4 4th Column 8 4th Row 1st Column 4 2nd Column 0 3rd Column 8 4th Column 16 EndLayout.

Select the correct answer below:

No, the associative property is not satisfied.

No, the identity and inverse properties are not satisfied.

No, none of the four properties are satisfied.

Yes, all four properties are satisfied.

To determine if the system defined in the table forms a group, we need to check the four properties of a group:

1. **Closure**: For every pair of elements \( a \) and \( b \) in the set, the result of the operation \( a \ast b \) must also be in the set.
2. **Associativity**: For every \( a, b, c \) in the set, \( (a \ast b) \ast c = a \ast (b \ast c) \).
3. **Identity element**: There must exist an element \( e \) in the set such that for every element \( a \), \( e \ast a = a \ast e = a \).
4. **Inverses**: For each element \( a \) in the set, there must exist an element \( b \) such that \( a \ast b = b \ast a = e \).

### Step 1: Define the table and elements
The operation table provided is:

| asterisk | 0 | 2 | 4 |
|----------|---|---|---|
| 0        | 0 | 0 | 0 |
| 2        | 0 | 4 | 8 |
| 4        | 0 | 8 | 16 |

### Step 2: Check Closure
- The results of the operation for all combinations of the elements (0, 2, 4) are in the set {0, 2, 4, 8, 16}.
- Closure is satisfied.

### Step 3: Check Associativity
To verify associativity, we would need to check all combinations, which can be tedious without clear patterns. However, we can test a few combinations:
- For example: \( (2 \ast 2) \ast 4 = 4 \ast 4 = 16 \) and \( 2 \ast (2 \ast 4) = 2 \ast 8 = 0 \).
- Since \( (2 \ast 2) \ast 4 \neq 2 \ast (2 \ast 4) \), associativity is **not satisfied**.

### Step 4: Check Identity
From the table, \( 0 \) appears to be an identity for some elements:
- \( 0 \ast 0 = 0 \), \( 0 \ast 2 = 0 \), \( 0 \ast 4 = 0 \).
- However, \( 2 \ast 0 \neq 2 \) and \( 4 \ast 0 \neq 4 \).
- Thus, there is **no identity element**.

### Step 5: Check Inverses
Since there is no identity element, we cannot find inverses.

### Conclusion
The system does not satisfy the associativity property, the identity property, and consequently, the inverse property.

The correct answer is:
**No, the identity and inverse properties are not satisfied.**

### Questions for Further Exploration
1. What examples of mathematical systems do satisfy all group properties?
2. Can a set be closed under an operation but still not form a group?
3. How can we demonstrate associativity using specific element combinations?
4. What modifications could be made to this operation to form a group?
5. How does the concept of a monoid differ from a group?

**Tip**: Always verify the properties of a group systematically to avoid missing critical aspects.

Given the mathematical system defined in the table below, determine if the system is a group.

Explore whether the given mathematical system meets the criteria for being a group. This solution involves checking closure, associativity, identity, and inverses based on the operation table provided. Ideal for Grades 10-12 students studying group theory.

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