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Complement and De Morgan's Laws — IISER Maths MCQs with Solutions
Free IISER Maths Complement and De Morgan's Laws MCQs with step-by-step solutions (16 questions). Part of Sets. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Complement and De Morgan's Laws · medium · theory
De Morgan's first law states that $(A \cup B)'$ equals:
A. $A \cap B$
B. $A' \cap B'$ ✓ Correct
C. $A' \cup B'$
D. $A' \cap B$
Solution: The complement of a union is the intersection of the complements.
Q2 — Complement and De Morgan's Laws · medium · theory
De Morgan's second law states that $(A \cap B)'$ equals:
A. $A' \cap B'$
B. $A \cup B'$
C. $A' \cup B'$ ✓ Correct
D. $A \cup B$
Solution: The complement of an intersection is the union of the complements.
Q3 — Complement and De Morgan's Laws · medium · numerical
If $U = \{1,2,\dots,10\}$, $A = \{1,2,3,4\}$, $B = \{3,4,5,6\}$, then $(A \cup B)'$ is:
A. $\{3,4\}$
B. $\{7,8,9,10\}$ ✓ Correct
C. $\{1,2,5,6\}$
D. $\{5,6,7,8\}$
Solution: $A\cup B = \{1,2,3,4,5,6\}$, so its complement is $\{7,8,9,10\}$.
Q4 — Complement and De Morgan's Laws · medium · numerical
With $U = \{1,2,\dots,10\}$, $A = \{1,2,3,4\}$, $B = \{3,4,5,6\}$, the set $(A \cap B)'$ is:
A. $\{1,2,5,6,7,8,9,10\}$ ✓ Correct
B. $\{7,8,9,10\}$
C. $\{3,4\}$
D. $\{1,2,5,6\}$
Solution: $A\cap B = \{3,4\}$, so its complement is everything else: $\{1,2,5,6,7,8,9,10\}$.
Q5 — Complement and De Morgan's Laws · medium · theory
For any set $A$, the expression $A \cap A'$ equals:
A. $\emptyset$ ✓ Correct
B. $A$
C. $U$
D. $A'$
Solution: A set and its complement are disjoint: $A\cap A' = \emptyset$.
Q6 — Complement and De Morgan's Laws · medium · theory
For any set $A$, the expression $A \cup A'$ equals:
A. $A$
B. $A'$
C. $\emptyset$
D. $U$ ✓ Correct
Solution: A set together with its complement is the whole universal set: $A\cup A' = U$.
Q7 — Complement and De Morgan's Laws · medium · numerical
If $n(U) = 50$ and $n(A) = 18$, then $n(A')$ is:
A. $50$
B. $32$ ✓ Correct
C. $18$
D. $68$
Solution: $n(A') = n(U) - n(A) = 50 - 18 = 32$.
Q8 — Complement and De Morgan's Laws · medium · theory
The identity $A - B = A \cap B'$ is a consequence of the definition of:
A. the empty set
B. complement and difference ✓ Correct
C. the power set
D. union only
Solution: Elements of $A$ not in $B$ are exactly those in $A$ and in $B'$.
Q9 — Complement and De Morgan's Laws · medium · numerical
If $U = \{x \in \mathbb{N} : x \le 10\}$ and $A = \{2,4,6,8,10\}$, $B = \{1,2,3,4,5\}$, then $A' \cap B'$ equals:
A. $\{7, 9\}$ ✓ Correct
B. $\{6, 8, 10\}$
C. $\{1, 3, 5\}$
D. $\emptyset$
Solution: By De Morgan, $A'\cap B' = (A\cup B)'$; $A\cup B=\{1,2,3,4,5,6,8,10\}$, complement $=\{7,9\}$.
Q10 — Complement and De Morgan's Laws · medium · numerical
If $n(U) = 120$, $n(A) = 60$, $n(B) = 45$, $n(A \cap B) = 25$, then $n(A' \cap B')$ is:
A. $60$
B. $80$
C. $40$ ✓ Correct
D. $35$
Solution: $n(A\cup B) = 60+45-25 = 80$; $n(A'\cap B') = 120 - 80 = 40$.
Q11 — Complement and De Morgan's Laws · medium · theory
The complement operation together with union and intersection satisfies $(A \cap B)' = A' \cup B'$. This is known as:
A. De Morgan's law ✓ Correct
B. the idempotent law
C. the associative law
D. the distributive law
Solution: This is one of De Morgan's two laws relating complement to union and intersection.
Q12 — Complement and De Morgan's Laws · hard · theory
For any set $A$, $A \cup (A' \cap B)$ simplifies to:
A. $A \cup B$ ✓ Correct
B. $A \cap B$
C. $A$
D. $B$
Solution: By distributivity: $A\cup(A'\cap B) = (A\cup A')\cap(A\cup B) = U\cap(A\cup B) = A\cup B$.
Q13 — Complement and De Morgan's Laws · hard · numerical
If $A$ and $B$ are subsets of $U$ with $n(U)=50$, $n(A')=30$, $n(B')=35$, and $n(A'\cap B')=15$, then $n(A\cap B)$ is:
A. $5$
B. $10$
C. $0$ ✓ Correct
D. $15$
Solution: $n(A)=20$, $n(B)=15$; $n(A\cup B)=n(U)-n(A'\cap B')=50-15=35$; $n(A\cap B)=20+15-35=0$.
Q14 — Complement and De Morgan's Laws · medium · theory
If $A \subseteq B$, then $B' \subseteq$ ?
A. $B$
B. $A'$ ✓ Correct
C. $U$
D. $A$
Solution: Taking complements reverses inclusion: $A\subseteq B \Rightarrow B'\subseteq A'$.
Q15 — Complement and De Morgan's Laws · hard · theory
The expression $(A \cap B') \cup (A \cap B)$ simplifies to:
A. $B$
B. $\emptyset$
C. $A$ ✓ Correct
D. $A \cup B$
Solution: $(A\cap B')\cup(A\cap B) = A\cap(B'\cup B) = A\cap U = A$.
Q16 — Complement and De Morgan's Laws · medium · numerical
If $n(U) = 200$, $n(A) = 120$, $n(B) = 90$ and $n(A \cup B) = 180$, then $n(A' \cap B')$ is:
A. $80$
B. $10$
C. $30$
D. $20$ ✓ Correct
Solution: $n(A'\cap B') = n((A\cup B)') = 200 - 180 = 20$.