Keep in mind, this is not the only order in which these can be simplified. Part 2 A. A*B*C+A*B+C = A*B+C ABC+AB+C = ABC+AB1+C Identity ABC+AB1+C = AB(C+1)+C Distribution AB(C+1)+C = AB(1)+C Null AB(1)+C = AB+C Identity B. A*(B+A+C) = A A(B+A+C) = AB+AA+AC Distribution AB+AA+AC = AB+A+AC Itempotent AB+A+AC = AB+1A+AC Identity AB+1A+AC = A(B+1+C) Distribution A(B+1+C) = A(1) Null A(1) = A Identity C. A*C+C*(!A+A*B) = C AC+C(!A+AB) = AC+!AC+ABC Distribution AC+!AC+ABC = C(A+!A)+ABC Distribution C(A+!A)+ABC = C(1)+ABC Inverse C(1)+ABC = C(1+AB) Distribution C(1+AB) = C(1) Null C(1) = C Identity D. (A+B*C*D+E)*E*D = D*E (A+BCD+E)ED = ADE+BCDEDE+DEE Distribution ADE+BCDEDE+DEE = ADE+BCDE+DE Itempotent ADE+BCDE+DE = ADE+BCDE+1DE Identity ADE+BCDE+1DE = (A+BC+1)DE Distribution (A+BC+1)DE = (1)DE Null 1DE = DE Identity E. F=(A+B)(A+C)(!A+C) = (A+B)*C (A+B)(A+C)(!A+C) = (A+B)(A!A+C!A+AC+CC) Distribution (A+B)(A!A+C!A+AC+CC) = (A+B)(0+C!A+AC+CC) Inverse (A+B)(0+C!A+AC+CC) = (A+B)(C!A+AC+CC) Identity (A+B)(C!A+AC+CC) = (A+B)(C!A+AC+C) Itempotent (A+B)(C!A+AC+C) = (A+B)(C!A+AC+1C) Identity (A+B)(C!A+AC+1C) = (A+B)(!A+A+1)C Distribution (A+B)(!A+A+1)C = (A+B)(1)C Null (A+B)(1)C = (A+B)c Identity Part 3 A. X+!X+Y x y x+!x+y 0 0 0+1+0 1 0 1 0+1+1 1 1 0 1+0+0 1 1 1 1+0+1 1 x+!x+y = 1+y Inverse 1+y = 1 Null B. X*(!X+Y) x y x*(!x+y) 0 0 0*(1+0) 0 0 1 0*(1+1) 0 1 0 1*(0+0) 0 1 1 1*(0+1) 1 x*(!x+y) = x*!x+x*y Distribution x*!x+x*y = 0+x*y Inverse 0+x*y = x*y Identity C. X*(!X*Y) x y x*(!x*y) 0 0 0*(1*0) 0 0 1 0*(1*1) 0 1 0 1*(0*0) 0 1 1 1*(0*1) 0 x*(!x*y) = (x*!x)*y Associative (x*!x)*y = 0*y Inverse 0*y = 0 Null D. !X*(X+!Y)x y !x*(x+!y) 0 0 1*(0+1) 1 0 1 1*(0+0) 0 1 0 0*(1+1) 0 1 1 0*(1+0) 0 !x*(x+!y) = !x*x+!x*!y Distribution !x*x+!x*!y = 0+!x*!y Inverse 0+!x*!y = !x*!y Identity !x*!y = !(x+y) DeMorgan E. !X+X*Y Use substitute A for !X, B for X, C for Y and use distribution. Then substitute back and simplify. Substitution is not required but it makes the opportunity to use distribution obvious. !x+x*y = a+b*c Substitution a+b*c = (a+b)*(a+c) Distribution (a+b)*(a+c) = (!x+x)*(!x+y) Substitute back (!x+x)*(!x+y) = (1)*(!x+y) Inverse 1*(!x+y) = !x+y Identity x y !x+x*y !x+y 0 0 1+0*0 1 1+0 1 0 1 1+0*1 1 1+1 1 1 0 0+1*0 0 0+0 0 1 1 0+1+1 1 0+1 1 f. !(!X*!X+Y) !(!x+!x+y) = !(!x+y) Itempotent !(!x+y) = !!x*!y DeMorgan !!x*!y = x*!y Cancellation x y !(!x*!x+y) x*!y 0 0 !(1*1+0) 0 0*1 0 0 1 !(1*1+1) 0 0*0 0 1 0 !(0*0+0) 1 1*1 1 1 1 !(0*0+1) 0 1*0 0 Part 4 - More DeMorgan 5 Points each A. C*(A+B+!(!C+!D))*D C*(A+B+!(!C+!D))*D = C*D(A+B+!!C*!!D) DeMorgan C*D(A+B+!!C*!!D) = C*D(A+B+C*D) Cancellation C*D(A+B+C*D) = C*D*A+C*D*B+C*D*C*D Distribution C*D*A+C*D*B+C*D*C*D = C*D*A+C*D*B+C*D Itempotent C*D*A+C*D*B+C*D = C*D*A+C*D*B+1*C*D Identity C*D*A+C*D*B+1*C*D = C*D(A+B+1) Distribution C*D(A+B+1) = C*D(1) Null C*D*1 = C*D Identity B. !((A*B+B*!C)*A) !((A*B+B*!C)*A) = !(A*A*B+A*B*!C) Distribution !(A*A*B+A*B*!C) = !(A*B+A*B*!C) Itempotent !(A*B+A*B*!C) = !(1*A*B+A*B*!C) Identity !(1*A*B+A*B*!C) = !(A*B(1+!C)) Distribution !(A*B(1+!C)) = !(A*B(1)) Null !(A*B(1)) = !(A*B) Identity