Equal Sets — IISER Maths MCQs with Solutions
Free IISER Maths Equal Sets MCQs with step-by-step solutions (13 questions). Part of Sets. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Equal Sets · easy · theory
Which pair of sets is equal?
A. $\{1, 2, 3\}$ and $\{1, 2, 4\}$
B. $\{1, 2\}$ and $\{1, 2, 3\}$
C. $\{1, 2, 3\}$ and $\{3, 2, 1\}$ ✓ Correct
D. $\{1\}$ and $\{1, 1, 1\}$ only by counting
Solution: Two sets are equal iff they have exactly the same elements; order and repetition do not matter, so $\{1,2,3\}=\{3,2,1\}$.
Q2 — Equal Sets · medium · theory
The sets $A = \{x : x^2 = 4\}$ and $B = \{2, -2\}$ are:
A. not equal but equivalent
B. such that $A \subset B$ strictly
C. disjoint
D. equal ✓ Correct
Solution: $x^2=4 \Rightarrow x=\pm2$, so $A=\{2,-2\}=B$: the sets are equal.
Q3 — Equal Sets · medium · theory
Two sets $A$ and $B$ are said to be equivalent if:
A. they have exactly the same elements
B. they have the same number of elements (same cardinality) ✓ Correct
C. one is a subset of the other
D. they are both empty
Solution: Equivalent sets have equal cardinality; equal sets have identical elements. Equal $\Rightarrow$ equivalent, but not conversely.
Q4 — Equal Sets · medium · theory
Which pair is equivalent but NOT equal?
A. $\{a, b, c\}$ and $\{1, 2, 3\}$ ✓ Correct
B. $\{x : x^2 = 1\}$ and $\{-1, 1\}$
C. $\{1, 2\}$ and $\{1, 2, 3\}$
D. $\{1, 2, 3\}$ and $\{3, 1, 2\}$
Solution: Both have three elements (equivalent) but different members (not equal).
Q5 — Equal Sets · medium · theory
If $A = \{x \in \mathbb{N} : x^2 - 3x + 2 = 0\}$, then $A$ equals:
A. $\{1, 2, 3\}$
B. $\{-1, -2\}$
C. $\{1, 2\}$ ✓ Correct
D. $\{2, 3\}$
Solution: $x^2-3x+2=(x-1)(x-2)=0 \Rightarrow x=1,2$, both natural.
Q6 — Equal Sets · medium · theory
The set $\{x : x \text{ is a letter in "FOLLOW"}\}$ equals:
A. $\{F, O, L, L, O, W\}$
B. $\{F, O, W\}$
C. $\{F, O, L, W\}$ ✓ Correct
D. $\{F, L, O, O, W\}$
Solution: Distinct letters of FOLLOW are F, O, L, W (repeats listed once).
Q7 — Equal Sets · hard · theory
If $\{a, b\} = \{2, 3\}$ and $a < b$, then $2a + b$ equals:
A. $6$
B. $8$
C. $5$
D. $7$ ✓ Correct
Solution: From $a<b$: $a=2, b=3$, so $2a+b = 4+3 = 7$.
Q8 — Equal Sets · medium · theory
Which statement is FALSE?
A. $\{1, 2, 3\}$ and $\{1, \{2, 3\}\}$ are equal ✓ Correct
B. A set does not change if its elements are reordered
C. Repetition of elements does not create a new set
D. $\{x : x^2 = 9\} = \{-3, 3\}$
Solution: $\{1,2,3\}$ has three number-elements, while $\{1,\{2,3\}\}$ has two elements (1 and the set $\{2,3\}$) — they are not equal.
Q9 — Equal Sets · medium · theory
If $A = \{x : x \text{ is a multiple of } 2\}$ and $B = \{x : x \text{ is an even integer}\}$, then:
A. $A = B$ ✓ Correct
B. $A \subset B$ strictly
C. $B \subset A$ strictly
D. $A$ and $B$ are disjoint
Solution: "Multiple of 2" and "even integer" describe exactly the same numbers, so $A=B$.
Q10 — Equal Sets · hard · theory
Sets $A$ and $B$ satisfy $A \subseteq B$ and $B \subseteq A$. Then:
A. $A \subset B$ strictly
B. nothing can be concluded
C. $A$ and $B$ are disjoint
D. $A = B$ ✓ Correct
Solution: Mutual inclusion $A\subseteq B$ and $B\subseteq A$ is precisely the definition of set equality: $A=B$.
Q11 — Equal Sets · hard · theory
If $\{2a + b,\ a - b\} = \{5,\ 1\}$ with $a - b = 1$, then the value of $a$ is:
A. $2$ ✓ Correct
B. $4$
C. $3$
D. $1$
Solution: From the pairing $a-b=1$ and $2a+b=5$: adding, $3a=6 \Rightarrow a=2$ (then $b=1$).
Q12 — Equal Sets · medium · theory
Two finite sets $A$ and $B$ have $n(A) = n(B)$. Then $A$ and $B$ are necessarily:
A. disjoint
B. equal
C. equivalent, but not necessarily equal ✓ Correct
D. subsets of each other
Solution: Same cardinality means equivalent; they need not have the same elements, so not necessarily equal.
Q13 — Equal Sets · medium · theory
The sets $A = \{x \in \mathbb{R} : x^2 = x\}$ and $B = \{0, 1\}$ are:
A. equivalent but not equal
B. equal ✓ Correct
C. such that $A \subset B$ strictly
D. disjoint
Solution: $x^2=x \Rightarrow x(x-1)=0 \Rightarrow x=0,1$, so $A = \{0,1\} = B$.