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Jamia Millia Islamia MCA Previous Year Questions (PYQs)

Jamia Millia Islamia MCA Sets And Relations PYQ


Jamia Millia Islamia MCA PYQ
Points within a set are connected by a line segment must follow the condition that points are





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2018 PYQ

Solution

In a convex set, any line joining two points lies entirely within the set.

Jamia Millia Islamia MCA PYQ
Order of the power set $ P(A) $ of a set $ A $ of order $ n $ is:





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2025 PYQ

Solution

If
$ |A| = n $

Then
$ |P(A)| = 2^n $

Number of subsets $ = 2^n $

Jamia Millia Islamia MCA PYQ
$\text{The set }(A\cap B)'\ \cup\ (B\cap C)\ \text{ is equal to:},$





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2021 PYQ

Solution

$ (A\cap B)'=A'\cup B' ,$ (De Morgan) $\Rightarrow (A\cap B)'\cup(B\cap C)=(A'\cup B')\cup(B\cap C)=A'\cup\big(B'\cup(B\cap C)\big)$ $B'\cup(B\cap C)=(B'\cup B)\cap(B'\cup C)=U\cap(B'\cup C)=B'\cup C$ $\Rightarrow A'\cup(B'\cup C)=A'\cup B'\cup C$

Jamia Millia Islamia MCA PYQ
In the group G = {2, 4, 6, 8} under multiplication modulo 10, the identity element is





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2018 PYQ

Solution

$(a × e) \bmod 10 = a.$ For $e=6$, $2×6=12\equiv2,\ 4×6=24\equiv4,\ 8×6=48\equiv8.$

Jamia Millia Islamia MCA PYQ
A partition of {1, 2, 3, 4, 5} is the family





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2018 PYQ

Solution

A partition divides a set into disjoint non-empty subsets covering all elements. {{1,2},{3,4,5}} satisfies it.

Jamia Millia Islamia MCA PYQ
Let P(S) denote the power set of set S. Which of the following is always TRUE?





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2018 PYQ

Solution

A partition divides a set into disjoint non-empty subsets covering all elements. {{1,2},{3,4,5}} satisfies it.

Jamia Millia Islamia MCA PYQ
G = {a, b, c} is an abelian group with ‘e’ as identity element. The order of other elements is A. 2, 2, 3 B. 3, 3, 3 C. 2, 2, 4 D. 2, 2, 2





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2018 PYQ

Solution

With 3 elements, it must be cyclic of order 3.

Jamia Millia Islamia MCA PYQ
The maximum number of equivalence relations on the set $A = \{1,2,3\}$ is:





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2021 PYQ

Solution

Number of equivalence relations = Number of partitions of set $A$. For 3 elements, Bell number $B_3 = 5$. $\boxed{\text{Answer: (D) 5}}$

Jamia Millia Islamia MCA PYQ
How many elements does the set $P\big({\varnothing, a, {a}, {{a}}}\big)$ has; where $a$ and $b$ are distinct elements and $P$ denotes the power set?





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2023 PYQ

Solution

The given set ${\varnothing, a, {a}, {{a}}}$ has $4$ elements.
Therefore,

P(S)=2S=24=16|P(S)| = 2^{|S|} = 2^4 = 16

Jamia Millia Islamia MCA PYQ
What is the cardinality of these sets in the order of their serial number? (i) ${a}$ (ii) ${{a}}$ (iii) ${a, {a}}$ (iv) ${a, {a}, {{a}}}$





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2023 PYQ

Solution

(i) ${a}$ → 1 element
(ii) ${{a}}$ → 1 element
(iii) ${a, {a}}$ → 2 elements
(iv) ${a, {a}, {{a}}}$ → 3 elements


Jamia Millia Islamia MCA PYQ
Suppose that $A_i = {1, 2, 3, \ldots, i}$ for $i = 1, 2, 3, \ldots$ Then find $\displaystyle \bigcup_{i=1}^{\infty} A_i = ?$ Here $Z$ denotes the set of integers.





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2023 PYQ

Solution

i=1Ai={1,2,3,}=Z+\bigcup_{i=1}^{\infty}A_i=\{1,2,3,\dots\}= \mathbb Z^{+} Z+\boxed{\mathbb Z^{+}}


Jamia Millia Islamia MCA PYQ
$,\text{Find } \displaystyle \bigcup_{i=1}^{\infty} A_i \text{ and } \bigcap_{i=1}^{\infty} A_i,\ \text{for every positive integer } i \text{ where } A_i={-i,i}.$ (Here $\mathbb Z$ denotes the set of integers.)





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2023 PYQ

Solution

$A_1={-1,1},\ A_2={-2,2},\ldots$ so $\displaystyle \bigcup_{i=1}^{\infty}A_i={\ldots,-3,-2,-1,1,2,3,\ldots}=\mathbb Z\setminus{0}$, $\displaystyle \bigcap_{i=1}^{\infty}A_i=\varnothing$.

Jamia Millia Islamia MCA PYQ
Which of the following relations are functions? (i) ${(1,(a,b)),\ (2,(b,c)),\ (3,(c,a)),\ (4,(a,b))}$ (ii) ${(1,(a,b)),\ (2,(b,a)),\ (3,(c,a)),\ (1,(a,c))}$ (iii) ${(1,(a,b)),\ (2,(a,b)),\ (3,(a,b))}$ (iv) ${(1,(a,b)),\ (2,(b,c)),\ (1,(c,a))}$





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2023 PYQ

Solution

A relation $R$ is a function iff every first component occurs exactly once. (i) first components $1,2,3,4$ appear once each $\Rightarrow$ function. (ii) $1$ appears twice $\Rightarrow$ not a function. (iii) $1,2,3$ appear once each $\Rightarrow$ function. (iv) $1$ appears twice $\Rightarrow$ not a function.

Jamia Millia Islamia MCA PYQ
$(A\cap B')\ \cup\ (A'\cap B)\ \cup\ (A'\cap B')$ is equal to





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2023 PYQ

Solution

$(A'\cap B)\cup(A'\cap B')=A'\cap(B\cup B')=A'$. Then $A'\cup(A\cap B')=(A'\cup A)\cap(A'\cup B')=\mathbf U\cap(A'\cup B')=A'\cup B'$.

Jamia Millia Islamia MCA PYQ
The number of students who take both the subjects’ mathematics and chemistry are 30. This represents 10% of the enrolment in mathematics and 12% of enrolment in chemistry. How many students take at least one of these two subjects?






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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2023 PYQ

Solution

Let total mathematics students be $M$, chemistry students be $C$, and both be $30$. \[ 0.1M=30 \Rightarrow M=300,\qquad 0.12C=30 \Rightarrow C=250. \] At least one $= M+C-\text{both}=300+250-30=520$.

Jamia Millia Islamia MCA PYQ
Let $A$ and $B$ be two disjoint subsets of a universal set $E$. Then $(A \cup B)\cap B'$ is equal to





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2017 PYQ

Solution

Since $A$ and $B$ are disjoint, elements common with $B'$ are only from $A$. Hence $(A \cup B)\cap B' = A.$

Jamia Millia Islamia MCA PYQ
$(A - B) - A$ is equal to





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2017 PYQ

Solution

$(A-B)$ means elements in $A$ but not in $B$. Subtracting $A$ again gives an empty set.

Jamia Millia Islamia MCA PYQ
Let 10 is the cardinality of set A. The number of bijective mappings from set A to itself is





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2017 PYQ

Solution

Bijections on a 10-element set = number of permutations = 10. So, 10 = 3628800.

Jamia Millia Islamia MCA PYQ
Let cardinality of set A and B are 2 and 5 respectively. The number of relations from A to B is





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2017 PYQ

Solution

$|A\times B| = 2\cdot5=10$. A relation is any subset of $A\times B$. Count = $2^{10}=1024$.

Jamia Millia Islamia MCA PYQ
Let cardinality of A and B are 3 and 10 respectively. The number of one-to-one functions from A to B is





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2017 PYQ

Solution

Injective maps from 3 to 10 = permutations $P(10,3)=10\cdot9\cdot8=720$.

Jamia Millia Islamia MCA PYQ
Let $A=\{1,2,3,4\}$ and $B=\{a,b\}$. The number of surjective mappings from A to B is





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2017 PYQ

Solution

Onto functions to a 2-element codomain: $2^{4}-\binom{2}{1}1^{4}=16-2=14$ (I.E. principle).

Jamia Millia Islamia MCA PYQ
Let R be a relation on the set of ordered pairs of positive integers such that ((p, q), (r, s)) ∈ R if and only if p − s = q − r. Which one of the following is true about R?





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Solution

Let R be a relation on ordered pairs of positive integers such that ((p,q),(r,s)) R iff p-s=q-r Check reflexive: For reflexive, take (p,q)=(r,s) Then condition becomes p-q=q-p 2p=2q p=q This is not true for every ordered pair, so R is not reflexive. Check symmetric: Given p-s=q-r Rearrange: p+r=q+s Now for symmetry we need r-q=s-p This also gives r+p=s+q, same condition. So R is symmetric.

Jamia Millia Islamia MCA PYQ
The relation represented by $R = \{(1,1), (2,2), (3,3), (1,3), (3,2), (1,2)\}$ on the set $A = \{1, 2, 3\}$ is what kind of relation?





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2020 PYQ

Solution

R includes $(1,1), (2,2), (3,3)$ → Reflexive. Check symmetry: $(1,2)$ exists but $(2,1)$ does not → Not symmetric. Check transitivity: $(1,2)$ and $(2,2)$ imply $(1,2)$ already → transitive holds. Hence, relation is reflexive and transitive but not symmetric.

Jamia Millia Islamia MCA PYQ
For $a, b \in \mathbb{R}$ define $a = b$ to mean that $|x| = |y|$. If $[x]$ is an equivalence relation in $R$, then the equivalence relation for $[17]$ is...





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2020 PYQ

Solution

Given relation $a = b \iff |a| = |b|$. Then equivalence class of $17$ is $[17] = \{x \in \mathbb{R} : |x| = |17|\} = \{-17, 17\}.$

Jamia Millia Islamia MCA PYQ
The sets A and B have the same cardinality if and only if there is a ..... correspondence from A to B.





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2020 PYQ

Solution

Two sets have the same cardinality if there exists a one-to-one and onto (bijective) correspondence between them.

Jamia Millia Islamia MCA PYQ
The relation $R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$ on $A=\{1,2,3\}$ is:





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2022 PYQ

Solution

Reflexive — yes (all $(a,a)$ are present). Symmetric — no, since $(1,2)\in R$ but $(2,1)\notin R$. Transitive — yes, because $(1,2)$ and $(2,3)$ imply $(1,3)$ (which exists).

Jamia Millia Islamia MCA PYQ
For real numbers $x,y$, define $x\,R\,y$ iff $x-y+\sqrt{2}$ is irrational. Then the relation $R$ is:





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2022 PYQ

Solution

For reflexivity: $xRx$ means $x-x+\sqrt{2}=\sqrt{2}$ (irrational) ⇒ true. For symmetry: $xRy⇒x-y+\sqrt{2}$ irrational, but $y-x+\sqrt{2}=-(x-y)+\sqrt{2}$ may be rational. Not always true ⇒ not symmetric. For transitivity: fails similarly.

Jamia Millia Islamia MCA PYQ
If $A$ be a set of cardinality $n$, then number of one-to-one onto functions from set $A$ to $A$ is …





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2016 PYQ

Solution

Number of bijective (one-one and onto) functions from a set of $n$ elements to itself $= n!$.

Jamia Millia Islamia MCA PYQ
If $R$ is the largest equivalence relation on a set $A$ and $S$ is any relation on $A$, then:





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2022 PYQ

Solution

Largest equivalence on $A$ is $A\times A$ (universal relation), so every relation $S$ on $A$ satisfies $S\subseteq A\times A=R$.

Jamia Millia Islamia MCA PYQ
If $f$ is a function from a finite set $A$ having $10$ elements to a finite set $B$ having $5$ elements, then number of functions from $A$ to $B$ is …





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2016 PYQ

Solution

Each of $10$ elements in $A$ can be mapped to any of $5$ elements in $B$. Hence total functions $= 5^{10}$.

Jamia Millia Islamia MCA PYQ
If $A$ and $B$ are two sets, then $(A \cup B)' \cap B$ is equal to …





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2016 PYQ

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 $\phi$.

Jamia Millia Islamia MCA PYQ
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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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2016 PYQ

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.

Jamia Millia Islamia MCA PYQ
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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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MCA 2016 PYQ

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.

Jamia Millia Islamia MCA PYQ
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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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2024 PYQ

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.}}$

Jamia Millia Islamia MCA PYQ
On A={1,2,3} let R={(1,2)}. Then R is 





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2024 PYQ

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 $(1,2)$∘$(2,\cdot)$.

Jamia Millia Islamia MCA PYQ
A={1,2,3}. Which $f:A\to A$ does not have an inverse?





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2024 PYQ

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.

Jamia Millia Islamia MCA PYQ
The maximum number of equivalence relations on the set $A = {1, 2, 3}$ are





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2024 PYQ

Solution

Number of equivalence relations = Number of partitions of the set = Bell number $B_3 = 5$.

Jamia Millia Islamia MCA PYQ
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


Jamia Millia Islamia MCA PYQ
If A={1,2,3} and B={4,5}. What is the number of elements in A×B ?





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2025 PYQ

Solution

 The number of elements in a Cartesian product is:
A× B= A× B    
Here,
|A| = 3
|B| = 2
So,
3  2 = 6

Jamia Millia Islamia MCA PYQ
Power set of empty set has exactly _____ subset.





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2018 PYQ

Solution

Power set of $\phi$ = ${\phi}$ has 1 subset.

Jamia Millia Islamia MCA PYQ
What is the cardinality of the power set of ${0,1,2}$?





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Jamia Millia Islamia MCA Previous Year PYQ Jamia Millia Islamia MCA JAMIA MILLIA ISLAMIA MCA 2018 PYQ

Solution

For a set of $n$ elements, power set size $= 2^n = 2^3 = 8.$


Jamia Millia Islamia MCA


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