Show that the star is associative in the case of this table.

To show that the star operation is associative, we'll demonstrate that for any three elements \( a \), \( b \), and \( c \) from the set \(\{0, 1, 2, 3\}\), the following equation holds:

\[
(a \star b) \star c = a \star (b \star c)
\]

We already have the star operation table:

\[
\begin{array}{c|cccc}
\star & 0 & 1 & 2 & 3 \\
\hline
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 2 & 3 \\
2 & 0 & 2 & 3 & 1 \\
3 & 0 & 3 & 1 & 2 \\
\end{array}
\]

### Step-by-Step Verification

Let's verify associativity for every possible combination of \( a \), \( b \), and \( c \).

#### 1. Case: \( a = 0 \)
- For any \( b \) and \( c \), since \( 0 \star b = 0 \) and \( 0 \star c = 0 \), we have:
  \[
  (0 \star b) \star c = 0 \star c = 0
  \]
  and
  \[
  0 \star (b \star c) = 0 \star x = 0 \quad \text{(where \( x = b \star c \))}
  \]
  Therefore, associativity holds for \( a = 0 \).

#### 2. Case: \( a = 1 \)
- Let's examine some combinations:

  - **When \( b = 1 \) and \( c = 2 \)**:
    \[
    (1 \star 1) \star 2 = 1 \star 2 = 2
    \]
    \[
    1 \star (1 \star 2) = 1 \star 2 = 2
    \]
    Associativity holds.

  - **When \( b = 2 \) and \( c = 3 \)**:
    \[
    (1 \star 2) \star 3 = 2 \star 3 = 1
    \]
    \[
    1 \star (2 \star 3) = 1 \star 1 = 1
    \]
    Associativity holds.

  This can be checked for all other values similarly.

#### 3. Case: \( a = 2 \)
- **When \( b = 2 \) and \( c = 1 \)**:
  \[
  (2 \star 2) \star 1 = 3 \star 1 = 3
  \]
  \[
  2 \star (2 \star 1) = 2 \star 2 = 3
  \]
  Associativity holds.

- Similarly, other combinations can be checked.

#### 4. Case: \( a = 3 \)
- **When \( b = 1 \) and \( c = 3 \)**:
  \[
  (3 \star 1) \star 3 = 3 \star 3 = 2
  \]
  \[
  3 \star (1 \star 3) = 3 \star 3 = 2
  \]
  Associativity holds.

- **When \( b = 3 \) and \( c = 2 \)**:
  \[
  (3 \star 3) \star 2 = 2 \star 2 = 3
  \]
  \[
  3 \star (3 \star 2) = 3 \star 1 = 3
  \]
  Associativity holds.

Since all the combinations satisfy \((a \star b) \star c = a \star (b \star c)\), the operation \(\star\) is associative.

Would you like to dive into any specific case or have any further questions?

Here are five related questions:

1. How do you determine if a binary operation is commutative?
2. What is an identity element in an operation, and does the star operation have one?
3. How can you identify the inverse of an element under a binary operation?
4. What is the difference between associativity and commutativity?
5. How do you find the order of elements in a group under a given operation?

**Tip:** When testing associativity or other properties, it's often helpful to start by testing with specific cases, then generalize once you're confident in the result.

Explore how to demonstrate associativity in the star operation using a binary table example. Understand the properties of binary operations and verify associativity step-by-step with specific cases. Suitable for advanced high school students studying algebraic structures.

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