The simple form of my question is given the equation:
$$\frac{a}{b} = \frac{c}{d}$$
Does this imply that: $ a = c$ and $b = d$ (I simply pair the numerators to themselves, and denominators to themselves).
If this is true, is there any basis? If not true, any thoughts or insights on why not?
The 'somewhat complex' form of my question stems from the Bayes Theorem. Given the Bayes Theorem defined as: $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$ Then with shuffling of the elements, we have: $$\frac{P(A)}{P(B)} = \frac{P(A|B)}{P(B|A)}$$
Again, would this mean that: $P(A) = P(A|B)$ and $P(B) = P(B|A)$
I ask this question because I recently did an example math on conditional probability and noticed the above applied to my example math.