For each of the following assertions, say whether it is true or false and support your answer with arguments or counterexamples where appropriate.

a. P(A, B) = P(A)P(B).

b. P(A | B) = P(A)P(B).

c. P(A, B) = P(A)P(B) − P(A | B).

d. P(A, B, C) = P(A | B, C)P(B | C)P(C).

e. P(A, B, C) = P(C | A, B)P(A)P(B) .

f. P(A, B, C) = P(A | B)P(B | C)P(C).

g. P(A, B, C) = P(A | B, C)P(B | A, C)P(C | A, B).

h. P(A, B, C) = P(C, B | A)P(A).

i. P(A, B, C) = P(C | A, B)P(A, B).

j. P(A, B) = P c P(A | B, C = c)P(B | C = c)P(C = c).

k. If P(X, Y |Z) = P(X |Z)P(Y |Z) then X is independent of Y given Z. 

l. If P(X, Y, Z) = P(X, Z)P(Y ) then X is independent of Y given Z.

m. If P(X, Y, Z) = P(X)P(Z)P(Y ) then X is independent of Y given Z.