Q) Assume that:
All HIFS are PIFS
Some SIFS are RIFS
All ZIFS are SIFS
All PIFS are ZIFS
The Long MethodFirst determine the following thereoms:
All HIFS are PIFS
All PIFS are ZIFS
All ZIFS are SIFS
Therefore:
All HIFS are ZIFS
All HIFS are SIFS
Some SIFS are RIFS
Therefore:
Some HIFS are RIFS
Some PIFS are RIFS
That leaves us with these thereoms:
All HIFS are PIFS
All PIFS are ZIFS
All ZIFS are SIFS
All HIFS are ZIFS
All HIFS are SIFS
Some SIFS are RIFS
Some HIFS are RIFS
Some PIFS are RIFS
Now:
Some SIFS are RIFS
Some HIFS are RIFS
Some PIFS are RIFS
Therefore the reverse is true:
Some RIFS are SIFS
Some RIFS are HIFS
Some RIFS are PIFS
A: All ZIFS are RIFS
B: Some HIFS are ZIFS
C: All RIFS are SIFS
D: All HIFS are RIFS
E: Some ZIFS are PIFS
We can now eliminate answers B through D, because these all contradict our thereoms which leaves us with two potential answers, A and E.
All RIFS that are HIFS are also PIFS
All RIFS that are HIFS are also ZIFS
All RIFS that are HIFS are also SIFS
All RIFS that are PIFS are also ZIFS
All RIFS that are PIFS are also SIFS
All RIFS that are ZIFS are also SIFS
Therefore:
Some but not all PIFS are HIFS
Some but not all ZIFS are HIFS
Some but not all SIFS are HIFS
Some but not all ZIFS are PIFS
Some but not all SIFS are PIFS
Some but not all SIFS are ZIFS
Therefore A is wrong and E is right.
"Some but not all ZIFS are PIFS"
E: Some ZIFS are PIFS
Part 2: Doing this kind of problem quickly.As a matter of fact it is actually really intuitive that the answer would be E, because of the answers provided, and the nature of the relationships between each element. First of all let's rearrange the starting functions in chronological order.
All HIFS are PIFS
All PIFS are ZIFS
All ZIFS are SIFS
Some SIFS are RIFS
First of all, we know that it is impossible for All HIFS to also be PIFS, ZIFS and SIFS, because only some SIFS are RIFS and since all elements are contained in the set called RIFS, that would constitute a contradiction.
Second of all we know that the system is hierarchical: HIFS come before PIFS which come before SIFS which come before RIFS.
So we know from the start that it would be impossible for All PIFS to be HIFS or something like that, because it comes below it in the hierachy. We don't even need to prove that E is right because it is the only answer which conforms to the hierarchial modelling of the system. Because it says that only some of the lower tier elements will be higher tier elements.
Also re-reading the post the correct reasoning is given pretty cogently here:
The answer is E -
If all PIFS are ZIFS, then you can consider PIF as being a subset of ZIF, as in, if you had a PIF, it would definitely be a ZIF, but the ZIF would not necessarily be a PIF, because the statements given to us are not commutative, ie. "All PIFS are ZIFS" does not imply that all ZIFS are PIFS. Hence, we can state with certainty that some ZIFS are PIFS ie. E is correct.