Statements:
I. All tables are charts.
II. Some charts are graphs.
III. Some graphs are pies.
Conclusions:
I. No table is a graph.
II. Some pies are charts.
Logical Reasoning: Analyzing Statements and Conclusions
This problem requires us to analyze a set of statements and determine which of the given conclusions logically follow from them. This is a type of deductive reasoning problem often referred to as a syllogism.
Understanding the Statements
We are given three statements that we must assume to be true:
- Statement I: All tables are charts. (Symbolically: $\forall \text{ table, chart}$)
- Statement II: Some charts are graphs. (Symbolically: $\exists \text{ chart, graph such that chart} \cap \text{ graph} \neq \emptyset$)
- Statement III: Some graphs are pies. (Symbolically: $\exists \text{ graph, pie such that graph} \cap \text{ pie} \neq \emptyset$)
We need to assess if the given conclusions are necessarily true based only on these statements.
Evaluating Conclusion I: No table is a graph
The conclusion states: No table is a graph. (Symbolically: $\neg \exists \text{ table, graph such that table} \cap \text{ graph} \neq \emptyset$).
Let's analyze this using the statements:
- From Statement I, the set of all 'tables' is entirely contained within the set of 'charts'.
- From Statement II, there is an overlap between the set of 'charts' and the set of 'graphs'. Some entities are both charts and graphs.
However, the overlap between 'charts' and 'graphs' might involve charts that are *not* tables. It is possible that the subset of charts that are tables is completely separate from the subset of charts that are also graphs. Consider a scenario:
- Charts = {ChartA, ChartB, ChartC, ChartD}
- Tables = {ChartA, ChartB} (All tables are charts - satisfies Statement I)
- Graphs = {ChartC, ChartD, GraphX} (Some charts (ChartC, ChartD) are graphs - satisfies Statement II)
In this scenario, 'Tables' = {ChartA, ChartB} and 'Graphs' = {ChartC, ChartD, GraphX}. There is no overlap between tables and graphs. So, "No table is a graph" holds true here.
BUT, consider another scenario:
- Charts = {ChartA, ChartB, ChartC, ChartD}
- Tables = {ChartA, ChartB} (All tables are charts - satisfies Statement I)
- Graphs = {ChartA, ChartB, GraphX} (Some charts (ChartA, ChartB) are graphs - satisfies Statement II)
In this second scenario, 'Tables' = {ChartA, ChartB} and 'Graphs' = {ChartA, ChartB, GraphX}. Here, ChartA and ChartB are both tables and graphs. Therefore, the conclusion "No table is a graph" is false in this case.
Since we can find a valid scenario where the conclusion is false, Conclusion I does not logically follow from the statements with certainty.
Evaluating Conclusion II: Some pies are charts
The conclusion states: Some pies are charts. (Symbolically: $\exists \text{ pie, chart such that pie} \cap \text{ chart} \neq \emptyset$).
Let's analyze this using the statements:
- From Statement II, some charts are graphs.
- From Statement III, some graphs are pies.
This means there's an overlap between Charts and Graphs, and a separate overlap between Graphs and Pies. However, these two overlaps might be completely distinct within the 'Graphs' category. The graphs that are charts might not be the same graphs that are pies.
Consider a scenario:
- Charts = {ChartA, ChartB, ChartC}
- Graphs = {ChartA, GraphX, GraphY} (Some charts (ChartA) are graphs - satisfies Statement II)
- Pies = {GraphX, PieZ} (Some graphs (GraphX) are pies - satisfies Statement III)
In this scenario, 'Pies' = {GraphX, PieZ} and 'Charts' = {ChartA, ChartB, ChartC}. There is no overlap between pies and charts. So, "Some pies are charts" is false here.
Consider another scenario:
- Charts = {ChartA, ChartB, ChartC}
- Graphs = {ChartA, GraphX, GraphY} (Some charts (ChartA) are graphs - satisfies Statement II)
- Pies = {ChartA, GraphX, PieZ} (Some graphs (ChartA, GraphX) are pies - satisfies Statement III)
In this second scenario, 'Pies' = {ChartA, GraphX, PieZ} and 'Charts' = {ChartA, ChartB, ChartC}. Here, ChartA is both a pie and a chart. Therefore, the conclusion "Some pies are charts" is true in this case.
Since we found a valid scenario where Conclusion II is false, it does not logically follow from the statements with certainty.
Overall Verdict
Based on the analysis, neither Conclusion I nor Conclusion II necessarily follows from the given statements. It is possible to construct scenarios where the statements are true, but the conclusions are false.
Similar Questions
- In this question, three statements are given, followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follows/follow from the statements.
Statements:
I. Some tables are chairs.
II. All desks are chairs.
III. No chair is a fan.
Conclusions:
I. Some desks are tables.
II. Some fans are desks. - In this question, three statements are given, followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follows/follow from the statements.
Statements:
All boxes are squares.
All squares are rectangles.
All rectangles are figures.
Conclusions:
I. All boxes are figures.
II. All squares are figures. Three statements are given, followed by Two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
Some fractions are marts.
No mart is a stud.
All studs are frogs.
Conclusions:
I. Some fractions are definitely not studs.
II. Some fractions are definitely not marts.
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 calculators are lanterns.
Some lanterns are mobiles.
Conclusions:
I. All mobiles are calculators.
II. No mobile is a lantern.
III. Some mobiles are lanterns.
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 parrots are crows.
All crows are eagles.
Conclusions:
I. All parrots are eagles.
Il. Some crows are parrots.
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 from the statements.
Statements:
No certificate is a paper.
All pens are papers.
Conclusions:
I. No certificate is a pen.
II. No paper is a certificate.
III. Some papers are pens.
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 officers are clerks.
All clerks are workers.
Conclusions:
I. Some workers are officers.
Il. No worker is an officer.
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 chargers are phones.
No phone is a watch.
Conclusions:
I. All phones are chargers.
Il. No charger is a watch.
Ill. No watch is a phone.
Two statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All wallets are bags.
Some bags are envelopes.
Conclusions:
I. Some wallets are envelopes.
II. No wallet is an envelope.
III. No bag is an envelope.Two statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
Some boys are brave.
All boys are punctual.
Conclusions:
I. Some brave are boys.
II. Some punctual are boys.
III. Some brave are punctual.
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:
Some students are players.
All players are male.
Conclusions:
I. All males are players.
II. Some males are students.
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:
No creative is an employer.
All experts are creative.
All workers are experts.
Conclusions:
I. No employer is an expert.
II. No worker is an employer.
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 actors are choreographers.
All choreographers are producers.
Not a single producer is a director.
Conclusions:
I. Some actors are directors.
II. Not a single actor is a director.
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 :
1. All flowers are tulips.
2. No tulips are whites.
Conclusions :
I. All tulips are flowers.
II. Some tulips are whites.
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 rats are dogs.
Some rats are hens.
Conclusions:
I. Some rats are dogs.
II. Some hens are rats.
