Final answer:
The statement 'a' entails 'b' requires context, such as whether 'b' must be true whenever 'a' is true. Equivalency cannot be established without knowing additional information, like if a factor is non-zero. Moreover, dependence in probability shows a relationship where the probability of events is not simply the product of their individual probabilities.
Step-by-step explanation:
When analyzing equations or relationships between formulas, it is crucial to consider the underlying principles and properties that govern these mathematical statements. For instance, if we have two formulas, let's denote them as 'a' and 'b', there are specific rules that need to be followed to determine if one entails the other or if they are equivalent.
Considering the statement 'a' entails 'b', in a mathematical context, this would often mean that whenever 'a' is true, 'b' must also be true. However, without additional context or a specific definition of what 'a' and 'b' represent, we cannot assume they are equivalent. Generally in mathematics, particularly when discussing equations, if 'a' entails 'b', and we are given that A × F = B × F, we cannot necessarily conclude A = B without additional information that F is not zero. Similarly, the concept of dependence as in the statement regarding P(A AND B) indicates a relationship between two events in probability. If P(A AND B) does not equal P(A)P(B), this shows a dependency between events A and B.
Oftentimes in mathematics, equations or statements need to be examined and manipulated using algebraic properties or axioms to determine the truthfulness of a statement or the nature of a relationship such as commutativity (A + B = B + A). Conclusions about equality or dependence require careful consideration of these properties and the context in which they are applied.