Jamia Millia Islamia MCA Sets And Relations Previous Year Questions (PYQs) – Page 4 of 5
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If $A$ and $B$ are two sets, then $(A \cup B)' \cap B$ is equal to …
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Solution
$(A \cup B)'$ means elements not in $A$ or $B$.
Hence, $(A \cup B)' \cap B$ contains elements that are both in $B$ and not in $B$, i.e., none.
So result is $\phi$.
If $A = \{a, b, c\}$ and $B = \{a, b, d, e, f\}$ are two sets, then number of elements in $(A - B) \times (A \cap B)$ is …
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Solution
$A - B = \{c\}$ (only $c$ not in $B$).
$A \cap B = \{a, b\}$.
Then $(A - B) \times (A \cap B) = \{(c, a), (c, b)\}$ → 2 elements.
If $A$ and $B$ are two disjoint sets having $3$ and $5$ elements respectively, then power-set of $A \times (B - A)$ contains … elements.
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Solution
Since $A$ and $B$ are disjoint, $B - A = B$.
Then $A \times B$ has $3 \times 5 = 15$ ordered pairs.
Power-set of it will have $2^{15}$ elements.
Let A={1,2,3} and R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}. Then R is
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Solution
$A={1,2,3},; R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}$
Reflexive: $(1,1),(2,2),(3,3)\in R\Rightarrow$ reflexive.
Symmetric: $(1,2)\in R$ but $(2,1)\notin R\Rightarrow$ not symmetric.
Transitive: check the only nontrivial chain: $(1,2)$ and $(2,3)\Rightarrow (1,3)$, which is in $R$.
Pairs with $(x,x)$ keep the second pair; $(1,3)$ can only compose with $(3,3)$ giving $(1,3)$; $(2,3)$ with $(3,3)$ gives $(2,3)$. All required compositions are in $R$.
$\boxed{\text{$R$ is reflexive and transitive, but not symmetric.}}$
On A={1,2,3} let R={(1,2)}. Then R is
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Solution
Not reflexive (missing $(1,1),(2,2),(3,3)$). Not symmetric (since $(2,1)\notin R$). Transitive holds vacuously because there is no pair starting at $2$ to trigger $(1,2)$∘$(2,\cdot)$.
A={1,2,3}. Which $f:A\to A$ does not have an inverse?
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Solution
A, C, and D are bijections (permutations), hence invertible. In B, both $2$ and $3$ map to $1$ (not one-to-one), so not bijective $\Rightarrow$ no inverse.
The maximum number of equivalence relations on the set $A = {1, 2, 3}$ are
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Solution
Number of equivalence relations = Number of partitions of the set = Bell number $B_3 = 5$.
A relation $R$ in a set $A$ is called ________, if $(a_1, a_2) \in R$ implies $(a_2, a_1) \in R$, for all $a_1, a_2 \in A$.
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Solution
If A={1,2,3} and B={4,5}. What is the number of elements in A×B ?
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Solution
The number of elements in a Cartesian product is:
A× B= A× B
Here,
|A| = 3
|B| = 2
So,
3 2 = 6
Power set of empty set has exactly _____ subset.
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Solution
Power set of $\phi$ = ${\phi}$ has 1 subset.
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