Assume that a set M of m items need to be fairly divided among a set N of n agents, and each agent i has an additive valuation function vi$,$ where vi$($S$)=$ Pj in S vi$(\{$j$\}),$S $$ M$.$ Show that

a$.$ Envy$-$freeness $($EF$)$ implies proportionality $($Prop$)$

b$.$ Envy$-$freeness up to one item $($EF$1)$ implies proportionality up to one item $($Prop$1)$ c$.$ Prop does not imply EF

d$.$ Prop$1$ does not imply EF$1$