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?
Show Hint
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
Hide Solution
Verified By Collegedunia
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.
Was this answer helpful?
0
0
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
Hide Solution
Verified By Collegedunia
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.
Was this answer helpful?
0
0
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
Hide Solution
Verified By Collegedunia
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.
Was this answer helpful?
0
0
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
Hide Solution
Verified By Collegedunia
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.
Was this answer helpful?
0
0
Top Questions on Set Theory
- In a class of 150 students, 75 students chose physics, 111 students chose mathematics and 40 students chose chemistry. All students chose at least one of the three subjects and at least one student chose all three subjects. The number of students who chose both physics and chemistry is equal to the number of students who chose both chemistry and mathematics, and this is half the number of students who chose both physics and mathematics. The maximum possible number of students who chose physics but not mathematics, is
- In a survey of 250 people about three activities (Reading, Sports, Travel), the following data are given:
130 like Reading,
110 like Sports,
120 like Travel,
55 like both Reading and Sports,
50 like both Sports and Travel,
45 like both Reading and Travel,
25 like all three.
How many like only Travel?
- A group of 120 students attend at least one of three workshops: Data, Logic, and Verbal.
48 attend Data, 60 attend Logic, 50 attend Verbal.
20 attend both Data & Logic, 15 attend both Logic & Verbal, 12 attend both Data & Verbal, and 8 attend all three.
How many students attend exactly one workshop? - Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:
- JEE Main - 2025
- Mathematics
- Set Theory
- The number of non-empty equivalence relations on the set \(\{1,2,3\}\) is :
- JEE Main - 2025
- Mathematics
- Set Theory
View More Questions
Questions Asked in CAT exam
- In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$, there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$. Then, $k$ equals?
- CAT - 2025
- Number Systems
- The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:
- CAT - 2025
- Number Systems
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is:
- CAT - 2025
- Linear Inequalities
- The number of distinct integers $n$ for which $\log_{\left(\frac14\right)(n^2 - 7n + 11)>0$ is:}
- CAT - 2025
- Linear Inequalities
- If $a - 6b + 6c = 4$ and $6a + 3b - 3c = 50$, where $a, b$ and $c$ are real numbers, the value of $2a + 3b - 3c$ is:
- CAT - 2025
- Linear & Quadratic Equations
View More Questions