Functions Question 698
Question: Let a relation R in the set R of real numbers be defined as (a, b) $ \in R $ if and only if $ 1+ab>o $ for all $ a,b\in R $ . The relation R is
Options:
A) Reflexive and symmetric
B) Symmetric and transitive
C) Only transitive
D) An equivalence relation
Show Answer
Answer:
Correct Answer: A
Solution:
[a] we have, $ R={(a,b):1+ab>0,ab\in R} $ Let $ a\in R\therefore a^{2}\ge 0 $ or $ 1+a^{2}>0 $ or $ (a,a)\in R $
$ \therefore $ R is reflexive. Let $ (a,b)\in R,\Rightarrow 1+ab>0\Rightarrow 1+ba>0 $
$ \Rightarrow (b,a)\in R\therefore R $ is symmetric. $ ( 2,\frac{1}{3} )\in R $ because $ 1+2( \frac{1}{3} )=\frac{5}{3}>0 $ $ ( \frac{1}{3},-1 )\in R $ because $ 1+\frac{1}{3}(-1)=\frac{2}{3}>0 $ Now, $ (2,-1)\in R $ if $ 1+2(-1)=-1<0, $ which is not true.
$ \therefore $ R is not transitive.
BETA
