e) The expression is [a + (~a)bc + (~b)c]
a + [(~a)b + (~b)]c we can say that [(~a)b + (~b)] <--> [(~a) + (~b)] (Prove it by yourself:-)
a + [(~a) + (~b)]c
a + (~a)c + (~b)c we can say that [a + (~a)c] <--> [a + c]
a + c + (~b)c it is obvious that [c + (~b)c] <--> [c]
So [a + c] is the reduced expression.