Comprehension
A and Bare two sets (e.g. A = Mothers, B = Women). The elements that could belong to both the sets (e.g. women who are mothers) is given by the set C = A·B. The elements which could belong to either A or B, or both, is indicated by the set D = A∪B. A set that does not contain any elements is known as a null set represented by ϕ (e.g. if none of the women in the set B is a mother, then C = A·B is a null set, or C = ϕ). Let ‘V’ signify the set of all vertebrates, ‘M’ the set of all mammals, ‘D’ dogs, ‘F’ fish, ‘A’ alsatian and ‘P’, a dog named Pluto.
Question: 1
Given $X = M \cap D$ such that $X = D$. Which of the following is true?
Given $X = M \cap D$ such that $X = D$. Which of the following is true?
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When $A \cap B = A$, it means $A \subseteq B$.
Updated On: Aug 4, 2025
- All dogs are mammals
- Some dogs are mammals
- $X = \varphi$
- All mammals are dogs
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The Correct Option is A
Solution and Explanation
$X = M \cap D$ are those elements which are both mammals and dogs. Given $X = D$, it means all dogs are in $M$, i.e., all dogs are mammals.
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Question: 2
If $Y = F \cap (D \cup V)$ is not a null set, it implies that:
If $Y = F \cap (D \cup V)$ is not a null set, it implies that:
Show Hint
Check element meanings literally; in pure set terms, a non-empty intersection means some shared membership.
Updated On: Aug 4, 2025
- All fish are vertebrates
- All dogs are vertebrates
- Some fish are dogs
- None of these
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The Correct Option is C
Solution and Explanation
$F$ = fish, $D \cup V$ = dogs or vertebrates. $Y$ not null means some fish are either dogs or vertebrates. Since all fish are vertebrates by definition, but the question implies intersection has elements not trivial, the case “some fish are dogs” fits.
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Question: 3
If $Z = (P \cap D) \cup M$, then:
If $Z = (P \cap D) \cup M$, then:
Show Hint
Breaking intersections first, then unions, clarifies element inclusion in set logic.
Updated On: Aug 4, 2025
- The elements of $Z$ consist of Pluto, the dog, or any other mammal
- $Z$ implies any dog or mammal
- $Z$ implies Pluto or any dog that is a mammal
- $Z$ is a null set
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The Correct Option is A
Solution and Explanation
$P \cap D$ = Pluto if Pluto is a dog. Union with $M$ (all mammals) gives Pluto (dog) plus all mammals.
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Question: 4
If $P \cap A = \varphi$ and $P \cup A = D$, then which of the following is true?
If $P \cap A = \varphi$ and $P \cup A = D$, then which of the following is true?
Show Hint
Empty intersection means no overlap; union covering a set means both subsets combine to whole set.
Updated On: Aug 4, 2025
- Pluto and alsatians are dogs
- Pluto is an alsatian
- Pluto is not an alsatian
- $D$ is a null set
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The Correct Option is A
Solution and Explanation
$P \cap A = \varphi$ ⇒ Pluto is not an alsatian. $P \cup A = D$ ⇒ Pluto and alsatians together make up all dogs. Thus both are dogs.
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