Register

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

258 views
2 2 votes

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 the relation $R$ is

  1. reflexive but not symmetric.
  2. an equivalence relation.
  3. reflexive and symmetric but not transitive.
  4. transitive but not symmetric.

2 Answers

1 1 vote
$$
(a, b) R(c, d) \Rightarrow 3 a d-7 b c \in \text { even }
$$

For reflexive

$$
(a, b) R(a, b) \Rightarrow 3 a b-7 b a=-4 a b \in \text { even }
$$

For symmetric

$$
(a, b) R(c, d)
$$

$$
=3 a d-7 b c=\underbrace{(7 a d-3 b c)}_{\text {even }}+(-4 a d-4 b c)
$$

then

$$
(c, d) R(a, b)=3 b c-7 a d
$$

$\in$ even
Now check for transitive

$$
\begin{aligned}
& (a, b) R(c, d) \text { and }(c, d) R(e, f) \text { then }(a, b) R(e, f) \\
& 3 a d-7 b c=2 m \quad \Rightarrow(2,5) R(6,8) \text { and } \\
& 3 c f-7 e d=2 n \quad(6,8) R(9,4) \\
& \text { then } 3 a f-7 e b \neq \text { even } \notin(2,5) R(9,4)
\end{aligned}
$$
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

5 5 votes
3 3 answers
287
287 views
GO Classes asked Apr 11
287 views
GO Classes DPP | GATE CS | Discrete Mathematics| Set Theory
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...
6 6 votes
4 4 answers
354
354 views
GO Classes asked Apr 11
354 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
221
221 views
GO Classes asked Apr 11
221 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...