Question

Let $R$ be a relation defined in the set of natural numbers $N$ as $R=\left\{\right(x-y):x,y\in N,2x+y=41\}$. Which of the following is true?

• A. $R$ is reflexive
• B. $R$ is symmetric
• C. $R$ is transitive
• D. At least one is false

Answer 1

Answer: (D)

At least one is false

We have $R=\left\{\right(x,y):2x+y=41,x,y\in N\}$.

$2x+y=41\Rightarrow x=\frac{41-y}{2}$

$y=1\Rightarrow x=20\in N.\therefore (20,1)\in R$

$y=2\Rightarrow x=\frac{39}{2}\notin N\therefore (\frac{39}{2},2)\notin R$

Similarly, $(19,3),(18,5),(17,7),(16,9),(15,11),(14,13),(13,15),(12,17),(11,19),(10,21),(9,23),(8,25),(7,27),(6,29),(5,31),(4,33),(3,35),(2,37),(1,39)\in R$.

$R$ is not reflexive, because $1\in N$ and $(1,1)\notin R$.

$R$ is not symmetric, because $(20,1)\in R$ but $(1,20)\notin R$.

$R$ is not transitive, because $(19,3),(3,35)\in R$ but $(19,35)\notin R$

$\therefore$ The correct answer is $D$.