- Evaluate the left-hand side (LHS) of the equation: \((1 * 2) * (3 * 1) = 1 * 2 = 1\).
- Simplify the equation to \(1 = 2 * (3 * x)\).
- Test each value in the set \(M = \{1, 2, 3\}\) for \(x\):
- For \(x=1\), \(2 * (3 * 1) = 2 * 2 = 2
eq 1\).
- For \(x=2\), \(2 * (3 * 2) = 2 * 3 = 3
eq 1\).
- For \(x=3\), \(2 * (3 * 3) = 2 * 1 = 1\).
- The solution is \(\boxed{x = 3}\).
Explanation
- Analysis of the problem and evaluation of the left-hand side
The problem asks us to solve the equation \((1 * 2) * (3 * 1) = 2 * (3 * x)\) within the set \(M = \{1, 2, 3\}\) using the provided Cayley table for the binary operation \(*\). The table is defined as follows:
\[ \begin{array}{|c|c|c|c|} \ * & 1 & 2 & 3 \\ 1 & 3 & 1 & 2 \\ 2 & 1 & 2 & 3 \\ 3 & 2 & 3 & 1 \\ \end{array} \]
Our first task is to compute the left-hand side (LHS) of the equation, \((1 * 2) * (3 * 1)\).
- Simplifying the equation
From the table, we identify the values for the components of the LHS:
- First, we look for \(1 * 2\) in the table, where the row 1 and column 2 intersect: \(1 * 2 = 1\).
- Next, we look for \(3 * 1\) in the table, where the row 3 and column 1 intersect: \(3 * 1 = 2\).
Substituting these values back into the expression for the LHS gives:
\[(1 * 2) * (3 * 1) = 1 * 2\]
Looking at the table again for \(1 * 2\), we find that \(1 * 2 = 1\). Therefore, the left-hand side of the equation is equal to 1.
- Solving for x by testing elements of M
With the left-hand side determined, the equation \((1 * 2) * (3 * 1) = 2 * (3 * x)\) simplifies to:
\[1 = 2 * (3 * x)\]
Now, we must test each element of the set \(M = \{1, 2, 3\}\) to find which value of \(x\) satisfies this equation.
- Case \(x = 1\): Calculate \(3 * 1 = 2\), then \(2 * (3 * 1) = 2 * 2 = 2\). Since \(2
eq 1\), \(x = 1\) is not a solution.
- Case \(x = 2\): Calculate \(3 * 2 = 3\), then \(2 * (3 * 2) = 2 * 3 = 3\). Since \(3
eq 1\), \(x = 2\) is not a solution.
- Case \(x = 3\): Calculate \(3 * 3 = 1\), then \(2 * (3 * 3) = 2 * 1 = 1\). Since \(1 = 1\), \(x = 3\) is a solution.
- Conclusion
By testing all possible elements of the set \(M = \{1, 2, 3\}\), we have determined that only \(x = 3\) satisfies the equation \((1 * 2) * (3 * 1) = 2 * (3 * x)\). Thus, the solution set is \(\{3\}\).
Examples
Binary operations defined by tables are fundamental in abstract algebra and computer science, representing structures like groups or modular arithmetic. In real life, such tables can model state changes in simple systems. For instance, imagine a light switch with three settings (1, 2, 3) where the "effect" of flipping switches changes based on the current setting. By mapping these states to a table, we can predict the final state of a device after a sequence of operations, just as we solved for the unknown variable \(x\) in this problem.