Sets — Chapter Test | Class 11 IMO

Q1

The symmetric difference of two sets A and B, denoted by A Δ B, is mathematically equivalent to:

The symmetric difference is the set of elements in A or in B, but not in both. This can be expressed as (A ∪ B) − (A ∩ B) or (A − B) ∪ (B − A).

Q2

The number of subsets of an empty set is:

The only subset of ∅ is itself.

Q3

Which of the following is an example of an infinite set?

The multiples of 5 (5, 10, 15, 20...) continue indefinitely. Even though the number of animals is huge, it is finite.

Q4

According to De Morgan’s Laws, the complement of the union of two sets, (A ∪ B)', is equal to:

De Morgan's first law states that the complement of a union is the intersection of their complements: (A ∪ B)' = A' ∩ B'.

Q5

A stationery shop owner in Mumbai has sets of pens. Set A has 10 Cello pens, Set B has 10 Reynolds pens. Sets A and B are:

Since both sets have exactly 10 items, their cardinality is the same [n(A) = n(B) = 10]. However, the items (elements) are different. Thus, they are equivalent but not equal.

Q6

If A ⊂ B, then what is A ∪ B?

If A is a subset of B, all elements of A are already in B. Taking the union adds no new elements to B, so A ∪ B = B.

Q7

Which of the following is a finite set?

Though the number of leaves on a large Banyan tree is extremely large, it is a countable, finite number. The others are mathematically infinite.

Q8

The formula n(A ∪ B) = n(A) + n(B) is valid ONLY when:

The general formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). If A and B are disjoint, n(A ∩ B) = 0, reducing the formula to n(A ∪ B) = n(A) + n(B).

Q9

If n(A)=4 and n(B)=5, then maximum possible value of n(A∩B) is:

Intersection cannot exceed smaller set size.

Q10

The set of all prime numbers less than 20 is given by:

Prime numbers are numbers greater than 1 that have only two factors: 1 and themselves. 1 is not a prime number. Therefore, the set is {2, 3, 5, 7, 11, 13, 17, 19}.

Q11

If A = {x : x ∈ R, x² = 16} and B = {x : x ∈ Z, x² = 16}, then:

In set A, real roots of x² = 16 are 4 and −4. In set B, integer roots of x² = 16 are 4 and −4. Thus A = {4, −4} and B = {4, −4}. The sets are equal.

Q12

In a group of 50 students, 28 study Hindi, 22 study Sanskrit and 8 study both. How many study at least one language?

28+22−8=42.

Q13

Which of the following collections is a well-defined set?

The vowels in the English alphabet are precisely defined as {a, e, i, o, u}.

Q14

If A = {1, 2} and B = {1, 2, 3, 4}, then the relationship between A and B is best described as:

Every element of set A is present in set B. Therefore, A is a subset of B, written as A ⊂ B.

Q15

If A={1,2,3}, then which is not a subset of A?

4 is not an element of A.