Does every monotonic social welfare function satisfy the

Question

A social welfare function F is called monotonic if for every alternative a ∈ A, and every pair of strict preference profiles P N and QN satisfying

a>p₁ c ⇒ a>Q₁ c, ∀c≠a, ∀i ∈ N,

the following is also satisfied:
a>F(P)c ⇒  a>F(Q)c, ∀c≠a.

In words, if for each individual i the ranking of alternative a relative to the other alternatives is not lowered in moving from Pi to Qi, then its ranking is not lowered in society's ranking, according to the social welfare function, when moving from profile P N to profile QN .

Answer the following questions:

(a) Does every monotonic social welfare function satisfy the unanimity property? Justify your answer.

(b) Does every monotonic social welfare function satisfy the independence of irrelevant alternatives property? Justify your answer.

(c) Is every social welfare function satisfying the independence of irrelevant alternatives property monotonic? Justify your answer.