Statements:
All flies are insects.
All insects are lasers.
Conclusions:
(I) Some lasers are insects.
(II) All flies are lasers.
Analyzing the Logical Statements
This problem involves deductive reasoning based on given statements. We need to determine if the conclusions logically follow from the premises.
Understanding the Statements
Let's represent the statements using set theory concepts:
- Statement 1: All flies are insects. This means the set of 'flies' is a subset of the set of 'insects'. If something is a fly, it must be an insect. We can denote this as: $Flies \subseteq Insects$.
- Statement 2: All insects are lasers. This means the set of 'insects' is a subset of the set of 'lasers'. If something is an insect, it must be a laser. We can denote this as: $Insects \subseteq Lasers$.
Evaluating Conclusion (I): Some lasers are insects.
From Statement 2, we know that All insects are lasers ($Insects \subseteq Lasers$). In classical logic, universal affirmative statements ("All A are B") imply that the subject category (A) is not empty. Therefore, if there exist insects, and all of them are lasers, it necessarily follows that there are some things that are both lasers and insects. Thus, Conclusion (I) logically follows.
Evaluating Conclusion (II): All flies are lasers.
We have the following relationships:
- $Flies \subseteq Insects$ (from Statement 1)
- $Insects \subseteq Lasers$ (from Statement 2)
This implies a transitive relationship. If the set of flies is entirely contained within the set of insects, and the set of insects is entirely contained within the set of lasers, then the set of flies must also be entirely contained within the set of lasers.
Mathematically, if $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$. Applying this here:
Since $Flies \subseteq Insects$ and $Insects \subseteq Lasers$, it follows that $Flies \subseteq Lasers$.
This directly means All flies are lasers. Therefore, Conclusion (II) logically follows.
Final Determination
Both Conclusion (I) and Conclusion (II) are valid deductions from the given statements.
| Conclusion | Follows from Statements? |
|---|---|
| (I) Some lasers are insects. | Yes |
| (II) All flies are lasers. | Yes |
Since both conclusions logically follow, the correct option is the one stating that both conclusions are valid.
Important Questions from Syllogism (Notes)
Take the given statements to be true even if they seem to be at variance with commonly known facts. Then decide which of the given conclusions logically follow the given statements.
Statements :
0% chairs are tables.
All computers are chairs.
Some books are tables.
Conclusions :
I. Not a single table is a computer.
II. Some books are not chairs.
Directions: Study the following Bar-chart and the data provided to answer the questions that follow: Sales of 3-different companies selling T-Shirts for the years from 2006 to 2009 are given. How much percentage of sales decreased for Brand X T-shirt in 2009 compared with 2007?
Statements:
All herbivores are carnivores.
Some carnivores are omnivores.
Conclusions:
i) Some herbivores are omnivores.
ii) Some carnivores are herbivores.
Statements:
All sweets are snacks.
All snacks are crisps.
All crisps are tasty.
Conclusions:
i) All sweets are crisps.
ii) All snacks are tasty.
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 furs are rabbits.
All furs are plants.
Conclusions:
(I): Some rabbits are plants.
(II): All plants are furs.
