let suppose Y=f(A,B,C)
the expression for y=AB+ABC+BC
1)Implicant : for y the implicants are AB,ABC,BC basically in simple words implicant is 'for any SOP , the product term that implies our y is basically implicant ' i mean when your implicant would be 1 then y would be 1.
for given sop set any product term (AB or ABC or BC) 1 and it will make y=1.
2)Prime implicant : Implicants whose removal don't now imply y (mean their presence is prime)are called PI.
for y PI are AB,BC as if they are removed and here we lost y. ABC is not PI as if you remove it from y then still our y is not lost.
3)Essential PI : An EPI of a function y is one that cover a minterm of F , not covered by any other PI of y.
Minterm for given function are x'(y+y')(z+z') and y'z(x+x') and xz(y+y')
When you would solve given function using K-Map you will get f=x'+y'+z
For f if any one of (x' , y'z , xz) is 1 then it would make f=1 so these all are implicants.
as f= x'+y'+z and if we remove anyone of x' or y' or z then here we lost f so all three are PI.