- For the first problem, we evaluate the Cayley table expression by calculating \((1 \varnothing 2) = 3\) and \((4 \varnothing 1) = 4\), then computing \(3 \varnothing (3 \varnothing 4) = 3 \varnothing 4 = 4\).
- For the second problem, the identity element is \(e = -1\), and the inverse formula is \(a^{-1} = -a - 2\), giving \(3^{-1} = -5\) and \(4^{-1} = -6\).
- The final calculation for the second problem is \(E = (-5) * (-6) = -10\).
- The final answers are 4 and -10 respectively. \(\boxed{4, -10}\)
Explanation
- Evaluating the Cayley Table Operation
The first part involves an operation \(\varnothing\) defined on the set \(\{1, 2, 3, 4\}\). We are given the Cayley table to determine the outcome of each operation between two elements. We need to compute \(E = (1 \varnothing 2) \varnothing [3 \varnothing (4 \varnothing 1)]\).
First, let's identify the results from the table:
- For \((1 \varnothing 2)\), we look at row 1 and column 2, which gives \(3\).
- For \((4 \varnothing 1)\), we look at row 4 and column 1, which gives \(4\).
Now, substitute these into the inner parts of the expression:
- The expression becomes \(3 \varnothing (3 \varnothing 4)\).
- Looking at the table for \(3 \varnothing 4\), we find row 3, column 4, which is \(4\).
- Finally, \(E = 3 \varnothing 4\). Looking at row 3, column 4 again, we find that the result is \(4\). Thus, \(E = 4\).
- Evaluating the Operation * and Inverse Elements
The second part involves an operation \(*\) defined on \(\mathbb{Z}\) by \(a * b = a + b + 1\).
First, find the identity element \(e\):
An identity element \(e\) must satisfy \(a * e = a * e = a\).
Substituting the definition: \(a + e + 1 = a \implies e + 1 = 0 \implies e = -1\).
Next, find the inverse element \(a^{-1}\) such that \(a * a^{-1} = e\):
Substituting the identity element \(-1\):
\(a + a^{-1} + 1 = -1 \implies a^{-1} = -a - 2\).
Now calculate \(3^{-1}\) and \(4^{-1}\):
- \(3^{-1} = -(3) - 2 = -5\).
- \(4^{-1} = -(4) - 2 = -6\).
Finally, compute \(E = 3^{-1} * 4^{-1}\):
\(E = (-5) * (-6) = (-5) + (-6) + 1 = -11 + 1 = -10\).
Examples
Operations like these are foundational to abstract algebra. For example, in computer science, modular arithmetic or bitwise operations often rely on defined tables or laws of composition (like identity and inverse) to manage data safely. Just as we found the inverse of an element in a group to reverse an operation, encryption algorithms use inverse transformations to decode information. Mastering these structures allows one to understand how complex systems, from computer hardware logic gates to advanced cybersecurity protocols, are built from simple, consistent rules.