Question

The relation P defined from R to R as a P b $\iff$ 1 + ab > 0, for all a, b $ϵ$ R is

• A. reflexive only
• B. reflexive and symmetric only
• C. transitive only
• D. equivalence

Answer 1

The correct option is B reflexive and symmetric only

$Pb\iff 1+ab>0$

i)Reflexive: $a\times a$ for any real value except 0 is positive.Hence $0\times 0+1>1\Rightarrow a\times a+1>0$

ii)Symmetric: if $a\times b+1>0$ then $b\times a+1$ will also be $>0$

iii)Not Transitive: a=-2,b=0\Rightarrow -2\times 0+1>0$

$b=0,c=4\Rightarrow 0\times 4+1>0$

but a is not related to c $\because a=-2,c=4\Rightarrow -2\times 4+1<0$