Final answer:
The question requires a comparison of boolean equations to determine which are equivalent, but the actual equations are not provided. To solve this, one would simplify each equation and compare their structures. Since the specific equations are missing, we cannot perform a detailed comparison.
Step-by-step explanation:
To determine which two of the given boolean equations are equivalent, one needs to analyze each equation and simplify them if possible, to see whether they reduce to the same expression. Unfortunately, the equations themselves are not provided in the prompt above, which makes it impossible to perform the actual comparison. However, in general, one would typically rearrange the terms in the boolean equations to isolate common variables and compare the structures. If they match after simplification, then the equations are equivalent and produce the same output. This process can be implemented using algebraic manipulation, such as factoring, distributing, and applying boolean identities.
If the question provided were related to the example about independent variables, we would need to reference probabilities and statistical dependence. However, without the context of the specific boolean equations, it is challenging to provide a detailed walkthrough on how to prove their equivalence.
In a scenario involving multiplication or division by the same number on both sides of an equation, as mentioned in the reference, applying the same operation to each term maintains the equality which is a fundamental principle in mathematics.