Register

GO Classes DPP | GATE CS | Discrete Mathematics| Set Theory

284 views
5 5 votes

Let $\mathrm{A}=\{1,2,3,4,5\}$. Let R be a relation on A defined by $x \mathrm{R} y$ if and only if $4 x \leq 5 \mathrm{y}$. Let m be the number of elements in R and n be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to R to make it a symmetric relation. Then $m+n$ is equal to :

  1. 23
  2. 26
  3. 25
  4. 24

3 Answers

2 2 votes
$$
\begin{aligned}
& A=\{1,2,3,4,5\} \\
& x R y \Leftrightarrow 4 x \leq 5 y \\
& 4 x \leq 5 y \quad \Rightarrow \quad \frac{x}{y} \leq \frac{5}{4} \quad \Rightarrow \frac{x}{y} \leq 1.25 \\
& R=\{(1,2),(1,3),(1,4),(1,5),(1,1),(2,2),(2,3),(2,4), \\
& (2,5),(3,3),(3,4),(3,5),(4,4),(4,5),(5,4),(5,5)\} \\
& \therefore \quad n(R)=m=16
\end{aligned}
$$

Elements to be added to $R$ to make it symmetric
$(1,2) \in R \quad \Rightarrow \quad(2,1)$ should be added
Similarly, $(3,1),(4,1),(5,1),(3,2),(4,2),(5,2),(4,3),(5,3)$
$\therefore 9$ elements should be added

$$
\begin{array}{ll}
\therefore & n=9 \\
\therefore & m+n=25
\end{array}
$$
Voting & Accepting an answer helps improve the community
Answer:
← Previous · GO Classes DPP | GAT...Next → · GO Classes DPP | GAT...
← Previous in category · GO Classes DPP | GAT...Next in category → · GO Classes DPP | GAT...
Position:
Show:

Related questions

2 2 votes
2 2 answers
255
255 views
GO Classes asked Apr 11
255 views
GO Classes DPP | GATE CS | Discrete Mathematics| Set Theory
Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b) R(c, d)$ if and only if $3 a d-7 b c$ is an even integer. Then ...
6 6 votes
4 4 answers
353
353 views
GO Classes asked Apr 11
353 views
GO Classes DPP | GATE CS | Discrete Mathematics| Set Theory
The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :7456
3 3 votes
2 2 answers
220
220 views
GO Classes asked Apr 11
220 views
GO Classes DPP | GATE CS | Discrete Mathematics| Set Theory
Let $A=\{1,2,3, \ldots, 10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m<n\right.$ and $\left.\operatorname{gcd}(m, n)=1\right\}$. Then $n(B)$ is equal to :29313736
3 3 votes
2 2 answers
238
238 views
GO Classes asked Apr 11
238 views
GO Classes DPP | GATE CS | Discrete Mathematics| Set Theory
Let $R=\{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equiva...