Syllogism

Let E = set of engineers, I = set of intelligent people, L = set of lazy people.
Statement 1: All engineers are intelligent. This means E is a subset of I (E I).
Statement 2: Some intelligent people are lazy. This means there is a non-empty intersection between I and L (I L ).
Conclusion I: Some engineers are lazy. (E L ?) From E I and I L .
Consider a Venn diagram. The set E is entirely within I. The set L overlaps with I.
The overlap between I and L could be entirely outside of E, or it could overlap with E.
Example:
Intelligent people = {i1, i2, i3, i4, i5}
Engineers = {i1, i2} (all engineers are intelligent)
Lazy people = {i3, i4} (some intelligent people are lazy; here i3, i4 are intelligent and lazy)
In this example, no engineer is lazy. So, "Some engineers are lazy" is not necessarily true. Conclusion I does not follow.
Conclusion II: All intelligent people are engineers. (I E?)
Statement 1 says E I. This means all engineers are intelligent, but it does not imply the converse that all intelligent people must be engineers. There can be intelligent people who are not engineers.
Example: Intelligent people = {i1, i2, i3}. Engineers = {i1, i2}. Here, i3 is intelligent but not an engineer.
So, Conclusion II does not follow.
Since neither Conclusion I nor Conclusion II necessarily follows from the statements, the correct option is (c).

Step 1: Analyze Statement 1
All actors are performers means every actor is within the performer group.
Step 2: Analyze Statement 2
Some performers are dancers means performers and dancers partially overlap.
Step 3: Check Conclusion I
"Some dancers are actors" — cannot be concluded for sure. Since dancers may or may not be actors.
Step 4: Check Conclusion II
"Some performers are actors" — this follows because all actors are performers, so definitely some performers are actors. Step 5: Conclusion
Only II follows.

Step 1: Given statements
All dogs are animals.
Some animals are cats.
Step 2: Analyze each option
(A) Some dogs are cats: Not necessarily true, since cats and dogs may be different subsets of animals.
(B) All cats are animals: Since some animals are cats, cats must be a subset of animals. Hence, true.
(C) Some animals are not dogs: This may or may not be true; we do not have full information.
(D) All animals are dogs: False, because cats are animals but not dogs.
Step 3: Conclusion
Only option (B) is definitely true.