I am trying to confirm these following statements:

(1) If is convex, then if and , then .

(2) Suppose . If is convex, then for any , . Basically this is saying if is indifferent to , then one would prefer over . I am not sure why this is the case.

A preference relation is monotone if for any and such that for . It is strongly monotone if for all and for some implies that

Source: www.ksg.harvard.edu/nhm/notes2006/notes3.pdf