Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All jams are bands.
All bands are sacks.
No sack is a can.
Conclusions:
(I) No jam is a can.
(II) Some sacks are jams.
This question requires us to analyze logical statements and determine which conclusions can be drawn from them. We need to check the validity of two conclusions based on the provided premises.
Analyzing Statements on Jams, Bands, Sacks, and Cans
Let's break down the given statements:
- Statement 1: All jams are bands.
This means the category 'jams' is entirely contained within the category 'bands'. We can represent this as: If something is a jam, then it is a band. Mathematically, if $J$ represents the set of jams and $B$ represents the set of bands, then $J \subseteq B$. - Statement 2: All bands are sacks.
Similarly, the category 'bands' is entirely contained within the category 'sacks'. If $S$ represents the set of sacks, this means $B \subseteq S$. - Statement 3: No sack is a can.
This indicates that the categories 'sacks' and 'cans' are mutually exclusive; they have no members in common. If $C$ represents the set of cans, then $S \cap C = \emptyset$.
Evaluating Conclusion I: No jam is a can
We need to determine if it's necessarily true that no jam is a can, given the statements.
- From Statement 1 ($J \subseteq B$) and Statement 2 ($B \subseteq S$), we can logically deduce that all jams are sacks ($J \subseteq S$). This is because if jams are within the band category, and the band category is within the sack category, then jams must be within the sack category.
- Statement 3 tells us that no sack is a can ($S \cap C = \emptyset$).
- Since all jams are a type of sack ($J \subseteq S$), and no sack can be a can, it logically follows that no jam can be a can. The set of jams is a subset of the set of sacks, which is disjoint from the set of cans. Therefore, the set of jams must also be disjoint from the set of cans ($J \cap C = \emptyset$).
- Thus, Conclusion (I) logically follows from the statements.
Evaluating Conclusion II: Some sacks are jams
We need to check if it's necessarily true that some sacks are jams.
- As established above, combining Statements 1 and 2 gives us $J \subseteq S$ (All jams are sacks).
- The statement "All jams are sacks" implies that if the set of jams is not empty, then there exist members within the set of sacks that are also jams. In standard logical reasoning problems, categories like 'jams' are assumed to exist unless stated otherwise.
- Therefore, there are indeed members of the set 'sacks' that are also members of the set 'jams'. This confirms that "Some sacks are jams".
- Thus, Conclusion (II) logically follows from the statements.
Final Verdict on Conclusions
Based on the analysis:
- Conclusion (I) logically follows from the premises.
- Conclusion (II) logically follows from the premises.
Therefore, both conclusions (I) and (II) are valid deductions from the given statements.
Important Questions from Syllogism
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
Some cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
