- Simplify the left-hand side \((1 * 2) * (3 * 1)\) using the provided Cayley table to obtain \(1\).
- Test each element of the set \(M = \{1, 2, 3\}\) in the right-hand side expression \(2 * (3 * x)\).
- Identify that for \(x = 3\), the expression evaluates to \(1\), satisfying the equation.
- The solution set is \(\boxed{\{3\}}\).
Explanation
- Analysis of the Problem
We are given a set \(M = \{1, 2, 3\}\) with a binary operation \(*\) defined by the following table:
\[ \begin{array}{|c|c|c|c|} \ * & 1 & 2 & 3 \\ 1 & 3 & 1 & 2 \\ 2 & 1 & 2 & 3 \\ 3 & 2 & 3 & 1 \\ \end{array} \]
Our objective is to solve the equation \((1 * 2) * (3 * 1) = 2 * (3 * x)\) for \(x \in \{1, 2, 3\}\). We proceed by evaluating the left-hand side and then testing each element for the right-hand side.
- Evaluating the Left-Hand Side
First, we calculate the left-hand side (LHS) of the equation: \((1 * 2) * (3 * 1)\).
Using the table:
- Look at the row for 1 and column for 2: \(1 * 2 = 1\).
- Look at the row for 3 and column for 1: \(3 * 1 = 2\).
Now, substitute these into the expression:
\((1 * 2) * (3 * 1) = 1 * 2\).
Referring back to the table, \(1 * 2 = 1\).
Thus, the LHS of the equation is 1.
- Solving for x
Now we must find \(x \in \{1, 2, 3\}\) such that \(2 * (3 * x) = 1\). We test each value of \(x\):
\(3 * 1 = 2\)
\(2 * 2 = 2\)
The result is 2, which does not equal 1.
\(3 * 2 = 3\)
\(2 * 3 = 3\)
The result is 3, which does not equal 1.
\(3 * 3 = 1\)
\(2 * 1 = 1\)
The result is 1, which matches the LHS.
Thus, the only solution is \(x = 3\).
Examples
In digital electronics and computing, binary operations on finite sets (like Boolean algebra or modular arithmetic) are used to define logic gates and state transitions. For instance, in a controller that has 3 specific states, defining how these states interact when a command is received helps ensure the system behaves predictably. Solving equations like this helps determine which input leads to a desired output state, ensuring the system functions as intended.