Condition A:
Given x, y in X such that $yRx$ then it follows that
$\lambda y +(1-\lambda)xRx$ for all $0< \lambda<1$
Condition B:
Given x, y in X such that $yPx$ then it follows that
$\lambda y +(1-\lambda)xPx$ for all $0< \lambda<1$
Show that the condition B implies the condition A.
R refers a weak preference relation and P is a strict preference relation.
I don’t understand how to show this implication.
What do you think? How can I show this? I am very confused.
A counterexample?
Suppose that $yPx$ iff $x=0$ and $y\ne 0$. Unless I'm missing something, $P$ is asymmetric and negatively transitive and a strict preference relation according to these slides. Moreover it satisfies condition B.
The associated weak preference relation $yRx$ iff $\neg xPy$, i.e. if $x=0$ or $y\ne 0$. Now $-1\,R\, 1$ but $\neg0\,R\,1$. Since $0=1/2*-1+1/2*1$ it would seem that condition A does not hold.