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These tables define the operations of the circle with a + inside and the star on a set of four elements: 0, 1, 2, and 3. For example 1 (circle with + inside) 3 = 2, because 2 is in the box to the right of 1 and below 3 in the circle with a + inside's table. Is the set closed with respect to the circle with the + inside. Is it closed with respect to the star?

To determine whether the set \(\{0, 1, 2, 3\}\) is closed under the operations defined by the tables for the circle with a plus inside (\(\oplus\)) and the star (\(\star\)), we need to check if the result of the operation between any two elements of the set always belongs to the set.

### Checking Closure for \(\oplus\) Operation:

Let's examine the operation table for \(\oplus\):

| \(\oplus\) | 0 | 1 | 2 | 3 |
|:-:|:-:|:-:|:-:|:-:|
| **0** | 0 | 1 | 2 | 3 |
| **1** | 1 | 0 | 3 | 2 |
| **2** | 2 | 3 | 0 | 1 |
| **3** | 3 | 2 | 1 | 0 |

We check each entry:

- Row 0: All results (0, 1, 2, 3) are in the set.
- Row 1: All results (1, 0, 3, 2) are in the set.
- Row 2: All results (2, 3, 0, 1) are in the set.
- Row 3: All results (3, 2, 1, 0) are in the set.

Since all the results of the \(\oplus\) operation are elements in the set \(\{0, 1, 2, 3\}\), the set is **closed** under the \(\oplus\) operation.

### Checking Closure for \(\star\) Operation:

Let's examine the operation table for \(\star\):

| \(\star\) | 0 | 1 | 2 | 3 |
|:-:|:-:|:-:|:-:|:-:|
| **0** | 0 | 0 | 0 | 0 |
| **1** | 0 | 1 | 2 | 3 |
| **2** | 0 | 2 | 3 | 1 |
| **3** | 0 | 3 | 1 | 2 |

We check each entry:

- Row 0: All results (0, 0, 0, 0) are in the set.
- Row 1: All results (0, 1, 2, 3) are in the set.
- Row 2: All results (0, 2, 3, 1) are in the set.
- Row 3: All results (0, 3, 1, 2) are in the set.

Since all the results of the \(\star\) operation are elements in the set \(\{0, 1, 2, 3\}\), the set is **closed** under the \(\star\) operation.

### Conclusion

- The set \(\{0, 1, 2, 3\}\) is **closed** with respect to the \(\oplus\) operation.
- The set \(\{0, 1, 2, 3\}\) is **closed** with respect to the \(\star\) operation.

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. What does it mean for a set to be closed under an operation?
2. How can we verify if a set is closed under a different operation?
3. What are other properties that operations on a set can have (e.g., associative, commutative)?
4. How can we determine if an operation has an identity element in the set?
5. What are examples of non-closed sets under certain operations?

**Tip:** Closure is a fundamental property when defining algebraic structures like groups, rings, and fields. Understanding closure helps in recognizing which operations and sets form valid algebraic systems.

Explore whether the set {0, 1, 2, 3} is closed under operations defined by the circle with a plus inside and star. Understand closure in algebraic structures with detailed operation tables and analysis. Suitable for high school students studying algebraic properties.

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