$1.(1$ point$)$ True or false? Justify your answer.

$($a$)$ If the core is nonempty, then the vector of Shapley values is always a core payoff vector.

$($b$)$ If the payoff vectors x and y belong to the core of a game G$,$ then for every $\backslash$alpha in $[0,1],$the payoff vector $\backslash$alpha x$+(1\backslash$alpha $)$y belongs to the core of G as well.

$($c$)$ The core of a cooperative game can be of size exactly two.