Three statements are followed by three conclusions numbered I, II and III. You have to consider these 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 given statements. Statements : Few chips are cakes. All cakes are ice creams. All ice creams are sweets. Conclusions : (I) Some sweets are cakes. (II) Some ice creams are cakes. (III) All cakes are sweets.
All conclusions I, II and III follow
Understanding Syllogism Statements and Conclusions
This question asks us to evaluate three conclusions based on three given statements. In syllogism problems, we must strictly adhere to the information provided in the statements, even if it contradicts common knowledge. We need to determine which conclusions logically follow from the statements.
Analyzing the Statements
Let's break down the given statements:
- Statement 1: Few chips are cakes. This means there is an overlap between the categories of "chips" and "cakes". In other words, some chips are cakes.
- Statement 2: All cakes are ice creams. This is a universal affirmative statement. It implies that the set of "cakes" is completely contained within the set of "ice creams".
- Statement 3: All ice creams are sweets. Another universal affirmative statement. This means the set of "ice creams" is completely contained within the set of "sweets".
Drawing Inferences from Statements
We can combine these statements to draw further inferences:
- From Statement 2 ("All cakes are ice creams") and Statement 3 ("All ice creams are sweets"), we can logically conclude that All cakes are sweets. This is a transitive property: if A is in B, and B is in C, then A is in C (where A=cakes, B=ice creams, C=sweets).
- From Statement 2 ("All cakes are ice creams"), if all cakes are ice creams, then it must be true that Some ice creams are cakes. The universal affirmative "All A are B" implies the particular affirmative "Some B are A".
- From the inference that "All cakes are sweets", it must be true that Some sweets are cakes. Again, applying the rule: "All A are B" implies "Some B are A".
- Statement 1 tells us "Few chips are cakes", which is equivalent to "Some chips are cakes". Combining this with "All cakes are ice creams", we can infer that Some chips are ice creams.
- Combining "Some chips are cakes" with "All cakes are sweets", we can infer that Some chips are sweets.
Evaluating the Conclusions
Now let's check the given conclusions against the statements and our inferences:
- Conclusion I: Some sweets are cakes.
As inferred from "All cakes are sweets" (which itself is derived from Statement 2 and Statement 3), this conclusion logically follows. If all members of one group (cakes) are part of another group (sweets), then some members of the larger group (sweets) must be from the first group (cakes). - Conclusion II: Some ice creams are cakes.
From Statement 2, "All cakes are ice creams", we can directly infer that "Some ice creams are cakes". This conclusion logically follows. - Conclusion III: All cakes are sweets.
As inferred from Statement 2 ("All cakes are ice creams") and Statement 3 ("All ice creams are sweets"), this conclusion logically follows.
Summary of Conclusions
Based on our analysis, all three conclusions logically follow from the given statements.
| Conclusion | Logical Follows? | Reasoning |
|---|---|---|
| I: Some sweets are cakes. | Yes | Derived from "All cakes are sweets" (Statements 2 & 3). |
| II: Some ice creams are cakes. | Yes | Derived directly from Statement 2 ("All cakes are ice creams"). |
| III: All cakes are sweets. | Yes | Derived from Statements 2 ("All cakes are ice creams") and 3 ("All ice creams are sweets"). |
Therefore, all conclusions I, II, and III follow.
Revision Table: Syllogism Rules
| Statement Type | Example | Valid Conversion |
|---|---|---|
| All A are B | All cakes are ice creams | Some B are A (Some ice creams are cakes) |
| No A are B | No dog is cat | No B are A (No cat is dog) Some A are not B Some B are not A |
| Some A are B | Some chips are cakes | Some B are A (Some cakes are chips) |
| Some A are not B | Some students are not smart | No valid direct conversion (Some B are not A is not always true) |
Additional Information: Types of Syllogism Statements
Categorical syllogisms use statements that relate categories or classes. The four standard types of categorical statements are:
- Universal Affirmative (A): All S are P (e.g., All dogs are mammals).
- Universal Negative (E): No S are P (e.g., No fish are birds).
- Particular Affirmative (I): Some S are P (e.g., Some students are athletes). 'Few' often translates to 'Some' in syllogism problems.
- Particular Negative (O): Some S are not P (e.g., Some fruits are not sweet).
Understanding these types helps in correctly interpreting the relationships between different categories mentioned in the statements.
Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A sphere of volume V is made of a material with lower density than water. While on Earth, it floats on water with its volume f1V (f1 < 1) submerged. On the other hand, on a spaceship accelerating with acceleration a < g (g is the acceleration due to gravity on Earth) in outer space, its submerged volume in water is f2V. Then:
A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
Consider the following statements:
1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
