Final answer:
The student's question involves sets and possibly vector operations, but due to unclear notation, a precise answer about set equality cannot be provided. In general, 'A × F = B × F' does not necessarily imply A equals B due to the properties of vector multiplication.
Step-by-step explanation:
The student seems to be asking about set equality and operations involving two sets A and B. However, the notation is not standard mathematical notation and seems to be incorrect or incomplete. Nevertheless, I can address the general concepts related to the questions posed.
For part (a), the statement 'A × F = B × F' indicates that if the cross product (vector product) of set A with another set or vector F equals the cross product of set B with F, it does not necessarily imply that A equals B. There could be instances where two different vectors, when crossed with a third vector, yield the same result due to the properties of vector multiplication.
The discussions in parts (b) and (c) likely relate to operations or relations involving A and B but without clearer context or notation, it's difficult to address precisely. As for the references to comparing areas A1, A2, and A3, or the discussion about the magnitudes of forces, these seem to require additional context to provide accurate explanations.
Overall, without the proper context or knowing the exact operations represented by 'FB F', 'FÃ', or 'BXA', I cannot provide a confident and accurate answer regarding whether these operations would imply A equals B.