A humanities course must discuss six out of eight topics-faith, knowledge, love, madness, revolution, skepticism, technology, and utopia-one at a time, each for one of six periods numbered consecutively from 1through 6. The ordering of topics must meet these conditions:
If faith is not discussed, utopia must be discussed last.
If technology is discussed, it must be discussed immediately before or else immediately after love. If faith is discussed, it must be discussed immediately before skepticism and immediately after madness. Knowledge or else revolution must be discussed first.
Which of the following is an acceptable sequence of topics discussed, in order from first through sixth?
Show Hint
- Knowledge, love, madness, faith, skepticism, technology
- Knowledge, madness, utopia, skepticism, technology
- Love, technology, revolution, madness, faith, skepticism
- Revolution, madness, faith, skepticism, love, technology
- Revolution, madness, skepticism, faith, technology, love
The Correct Option is D
Solution and Explanation
This question asks for a valid sequence. We must check each option against the four rules. The first option to satisfy all rules is the correct answer.
Step 2: Detailed Explanation:
- (A) K, L, M, F, S, T: Rule 4 (K or R first) is met. Rule 3 (MFS block) is met. But Rule 2 is violated: T is discussed but is not adjacent to L.
- (B) K, M, U, S, T: This sequence only has 5 topics. It is not a valid schedule.
- (C) L, T, R, M, F, S: Rule 4 is violated: L is first, but it must be K or R.
- (D) R, M, F, S, L, T: - Rule 4 (K or R first): Met, R is first. - Rule 3 (F \(\rightarrow\) MFS block): Met, we have the block MFS in periods 2-4. - Rule 2 (T \(\rightarrow\) LT/TL block): Met, we have the block LT in periods 5-6. - Rule 1 (\(\neg F \rightarrow U_6\)): Met, F is discussed, so the rule does not apply. All rules are satisfied. This is an acceptable sequence.
- (E) R, M, S, F, T, L: Rule 3 is violated: F is discussed but is not immediately preceded by M and followed by S. The order is S, F.
Step 3: Final Answer:
The sequence in option (D) is the only one that satisfies all the given conditions.
If exactly one topic is discussed between faith and love, that topic could be
Show Hint
- knowledge
- revolution
- skepticism
- technology
- utopia
The Correct Option is C
Solution and Explanation
This is a conditional question. We must add the new condition "F-X-L" or "L-X-F" to our existing rules and determine what X could be.
Step 2: Detailed Explanation:
1. The new condition requires a block of F-X-L or L-X-F.
2. Since F is discussed, Rule 3 must apply: we must have the MFS block.
3. In the MFS block, the topic immediately following F is S.
4. To satisfy the condition that exactly one topic is between F and L, that topic must be the one adjacent to F in the required block. In the MFS block, S is the topic adjacent to F.
5. Therefore, the sequence of topics must contain the larger block M-F-S-L. The topic (X) between F and L is S (skepticism).
6. We can confirm this is possible by constructing a full valid sequence. We have the 4-topic block M-F-S-L. We also know T must be discussed adjacent to L (Rule 2), giving us the 5-topic block M-F-S-L-T. To satisfy Rule 4, R or K must be first. A valid sequence is: R, M, F, S, L, T. In this sequence, the topic between F and L is indeed S.
Step 3: Final Answer:
Given the requirement of the MFS block, the only topic that can be immediately after F (and thus between F and L) is skepticism.
If neither faith nor madness is discussed and if revolution is discussed fourth, then skepticism must be discussed
Show Hint
- first
- second
- third
- fourth
- fifth
The Correct Option is
Solution and Explanation
This is a conditional question based on the logic game from the previous page. We apply the new conditions and deduce the fixed position of skepticism.
Initial Rules Recap: 1. \(\neg F \rightarrow U_6\) 2. T \(\rightarrow\) LT or TL block 3. F \(\rightarrow\) MFS block 4. \(K_1 \lor R_1\)
Step 2: Applying New Conditions and Deducing the Schedule:
1. New Conditions: Faith (F) and madness (M) are NOT discussed. Revolution (R) is discussed in period 4.
2. Topics Not Discussed: F and M. This means the 6 topics that ARE discussed are: K, L, R, S, T, U.
3. Rule 1: Since F is not discussed (\(\neg F\)), U must be discussed last (in period 6). Our schedule now looks like: \_ , \_ , \_ , R , \_ , U.
4. Rule 4: Knowledge (K) or revolution (R) must be first. Since R is in period 4, K must be in period 1. Our schedule becomes: K , \_ , \_ , R , \_ , U.
5. Topics placed so far: K, R, U. Remaining topics to place: L, S, T. Open slots: 2, 3, 5.
6. Rule 2: Technology (T), if discussed, must be next to Love (L). Both T and L must be placed in the remaining slots. For them to be adjacent, they must occupy slots 2 and 3. So, the (L,T) block is in periods 2 and 3. Our schedule is: K , (L/T) , (T/L) , R , \_ , U.
7. Final Placement: The only remaining topic is skepticism (S), and the only remaining open slot is period 5.
The complete schedule must be: K, (L/T), (T/L), R, S, U.
Step 3: Final Answer:
Based on the deductions, skepticism (S) must be discussed in the fifth period.
If revolution and utopia are the first two topics discussed, the two topics not discussed could be
Show Hint
- faith and love
- faith and technology
- knowledge and skepticism
- love and madness
- love and technology
The Correct Option is
Solution and Explanation
We add the new condition that R is 1st and U is 2nd. We then need to find a valid pair of topics that could be excluded from the 6-topic schedule.
Step 2: Detailed Explanation:
1. New Conditions: The schedule begins R, U, ...
2. Rule 4 (\(K_1 \lor R_1\)): This rule is satisfied. It does not mean K cannot be discussed at all, only that it cannot be first.
3. Rule 1 (\(\neg F \rightarrow U_6\)): The condition is that U is in period 2, not 6 (\(\neg U_6\)). By the contrapositive, F MUST be discussed.
4. Rule 3 (F \(\rightarrow\) MFS block): Since F must be discussed, the three-topic block MFS must be included in the schedule.
5. Schedule so far: R is in 1, U is in 2. The MFS block must be placed in three consecutive slots. The only available space for this is slots 3, 4, and 5. So the schedule is: R, U, M, F, S, \_.
6. Identifying the last topic: We have one slot left (period 6). The topics used are R, U, M, F, S. The remaining topics from the original pool of eight are K, L, T. We must choose one of these for the final slot.
7. Checking possibilities for slot 6: - Can slot 6 be T? If we place T in 6, the schedule is R, U, M, F, S, T. Rule 2 requires T to be adjacent to L. L is not in the schedule, so this is a violation. T cannot be the 6th topic. - Can slot 6 be L? If we place L in 6, the schedule is R, U, M, F, S, L. Rule 2 (T \(\rightarrow\) LT/TL) is not triggered because T is not discussed. All other rules are satisfied. This is a valid schedule. The topics NOT discussed are K and T. - Can slot 6 be K? If we place K in 6, the schedule is R, U, M, F, S, K. Rule 2 is not triggered because T is not discussed. All other rules are satisfied. This is also a valid schedule. The topics NOT discussed are L and T.
8. Conclusion: There are two possible pairs of topics that are not discussed: \{K, T\} and \{L, T\}. We check the options to see which of these pairs is listed.
Step 3: Final Answer:
The pair \{love, technology\} is a possible pair of undiscussed topics, as shown by the valid schedule R, U, M, F, S, K.
If knowledge is not discussed, the other topic not discussed could be
Show Hint
- faith
- love
- madness
- revolution
- skepticism
The Correct Option is A
Solution and Explanation
We are given that K is one of the two undiscussed topics. We must find what the second undiscussed topic could be by testing each option.
Step 2: Detailed Explanation:
1. Initial Deduction: If K is not discussed, then from Rule 4, R must be discussed first (\(R_1\)).
2. Test each option: We need to see if we can form a valid 6-topic schedule where the two "out" topics are K and the topic in the option.
- (A) faith: Can the "out" topics be \{K, F\}? - The 6 "in" topics are \{L, M, R, S, T, U\}. - We know \(R_1\). - Since F is out (\(\neg F\)), Rule 1 applies: U must be last (\(U_6\)). - Since F is out, Rule 3 does not apply. - T is in, so Rule 2 applies: L and T must be adjacent (LT or TL block). - We need to schedule \{M, S, L, T\} into slots 2, 3, 4, 5, with R in 1 and U in 6. The LT block must fit. - A valid schedule is possible: R, M, S, T, L, U. This schedule obeys all rules. - Therefore, it is possible for faith to be the other topic not discussed.
- (B) love: Can the "out" topics be \{K, L\}? - The "in" topics include T but not L. This violates Rule 2 (If T is discussed, L must be adjacent). So, this is not possible.
- (C) madness: Can the "out" topics be \{K, M\}? - The "in" topics include F but not M. This violates Rule 3 (If F is discussed, M must be immediately before it). So, this is not possible.
- (D) revolution: Can the "out" topics be \{K, R\}? - This violates Rule 4, which states that K or R must be discussed first. At least one must be in. So, this is not possible.
- (E) skepticism: Can the "out" topics be \{K, S\}? - The "in" topics include F but not S. This violates Rule 3 (If F is discussed, S must be immediately after it). So, this is not possible.
Step 3: Final Answer:
The only option that allows for a valid schedule is faith.
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