Question

We start with the set {1, 3, 6}. In each step, we may choose any two of these numbers, let's call them a and b, and replace them by 0.6a - 0.8b and 0.8a + 0.6b. Can we reach the set {2, 4, 5}?

A. Yes
B. No
C. It depends on the number of steps.
D. The set cannot be reached.

Answer 1

It is not possible to reach the set {2, 4, 5} from the set {1, 3, 6} using the given transformation because the transformation preserves the sum of numbers and the sum of the two sets are different.

Step-by-step explanation:

The question asks whether we can reach the set {2, 4, 5} starting from the set {1, 3, 6} using a specific transformation on any two numbers, where the transformation is defined by replacing two numbers a and b with 0.6a - 0.8b and 0.8a + 0.6b respectively.

To check for the possibility, we can analyze the original and the target sets. Notice that both sets have three elements, and consider the transformation properties. The transformation is linear and preserves the sum of the original numbers. That is:

• Sum of set {1, 3, 6} = 1 + 3 + 6 = 10
• Sum of set {2, 4, 5} = 2 + 4 + 5 = 11

Since the sum of the elements in the target set is different from the sum of the original set, and the transformation keeps the sum invariant (does not change it), it is impossible to reach the set {2, 4, 5} from the set {1, 3, 6} by applying the given transformation.

The correct answer to the question is therefore B. No, the set cannot be reached.