Jamia Millia Islamia MCA Sets And Relations Previous Year Questions (PYQs) – Page 3 of 5
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Showing 41 questions
Let $A=\{1,2,3,4\}$ and $B=\{a,b\}$. The number of surjective mappings from A to B is
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Solution
Onto functions to a 2-element codomain: $2^{4}-\binom{2}{1}1^{4}=16-2=14$ (I.E. principle).
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.
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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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.
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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Solution
Given relation $a = b \iff |a| = |b|$.
Then equivalence class of $17$ is $[17] = \{x \in \mathbb{R} : |x| = |17|\} = \{-17, 17\}.$
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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Solution
Two sets have the same cardinality if there exists a one-to-one and onto (bijective) correspondence between them.
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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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).
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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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.
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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Solution
Number of bijective (one-one and onto) functions from a set of $n$ elements to itself $= n!$.
If $R$ is the largest equivalence relation on a set $A$ and $S$ is any relation on $A$, then:
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Solution
Largest equivalence on $A$ is $A\times A$ (universal relation),
so every relation $S$ on $A$ satisfies $S\subseteq A\times A=R$.
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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Solution
Each of $10$ elements in $A$ can be mapped to any of $5$ elements in $B$.
Hence total functions $= 5^{10}$.
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