- For the first problem, calculating step-by-step: \(4 * 6 = 0\), \(2 * 5 = 4\), then \(0 * 4 = 8\), and finally \(8 * 3 = -18\).
- For the second problem, verifying statements:
- I. \(a * a = 2a\) is True.
- II. \(a * b = b * a\) is False.
- III. \((a^{-1}) * a = 0\) is True (assuming \(a^{-1} = -a\)).
- Final answer: \(\boxed{-18; I, III}\)
Explanation
- Solving Problem 1
We analyze the two given problems involving custom-defined operations.
Problem 1: The operation is defined as \(a * b = 2b - 3a\). We need to calculate \([(4 * 6) * (2 * 5)] * 3\).
First, we evaluate the inner operations:
- \(4 * 6 = 2(6) - 3(4) = 12 - 12 = 0\)
- \(2 * 5 = 2(5) - 3(2) = 10 - 6 = 4\)
Next, we substitute these results into the next part of the expression:
- \((4 * 6) * (2 * 5) = 0 * 4 = 2(4) - 3(0) = 8 - 0 = 8\)
Finally, we compute the result of this value operated with 3:
- \(8 * 3 = 2(3) - 3(8) = 6 - 24 = -18\)
Thus, \([(4 * 6) * (2 * 5)] * 3 = -18\).
- Solving Problem 2
Problem 2: The operation is defined as \(a * b = a^2 - b^2 + a + b\). We evaluate the three statements:
I. \(a * a = 2a\):
- \(a * a = a^2 - a^2 + a + a = 0 + 2a = 2a\). This statement is True.
II. \(a * b = b * a\):
- \(a * b = a^2 - b^2 + a + b\)
- \(b * a = b^2 - a^2 + b + a\)
- Comparing the two, they are equal only if \(a^2 - b^2 = b^2 - a^2\), which means \(2a^2 = 2b^2\), or \(a^2 = b^2\). Since this is not true for all \(a\) and \(b\), the statement is False.
III. \((a^{-1}) * a = 0\):
- If we interpret \(a^{-1}\) as the additive inverse \(-a\), we test:
\((-a) * a = (-a)^2 - a^2 + (-a) + a = a^2 - a^2 - a + a = 0\).
Under this standard interpretation of inverse in algebraic structures related to additive groups, the statement is True.
Therefore, statements I and III are true.
- Final Result
We have completed both problems. The result of the first calculation is \(-18\), and the true statements for the second problem are I and III. \(\boxed{-18; I, III}\).
Examples
Custom operations like the ones defined here are common in fields like cryptography, where complex functions are applied to data to ensure security. By defining a unique "key" operation, we can create algorithms that transform inputs in specific, reversible, or non-reversible ways. Understanding how to systematically apply these rules is foundational for computer science and advanced mathematics.