## EXAMPLES FOR WEEK 20

The following examples are related to the work covered in WEEK 20.

- Simplify A.(B + C) + A.NOT B
- Simplify A.B.NOT C + A.NOT B.NOT C
- Simplify X.Y + Y.Z + NOT X.Z + Y.NOT Z

First Example:- **Simplify A.(B + C) + A.NOT B**

"Multiply out" the first bracket.
A.B + A.C + A.NOT B
Collect like terms.
A.B + A.NOT B + A.C
Take out common factor.
A.(B + NOT B) + A.C
Use boolean algebra x + not x = 1
A.1 + A.C
Use boolean algebra x.1 = x
A + A.C
Take out common factor.
A.(1 + C)
Use boolean algebra 1 + x = 1
A.1
Use boolean algebra x.1 = x
A
CHECK USING TRUTH TABLES
C B A NOT B (B + C) A.(B + C) A.NOT B OUTPUT = A.(B + C) + A.NOT B
0 0 0 1 0 0 0 0
0 0 1 1 0 0 1 1
0 1 0 0 1 0 0 0
0 1 1 0 1 1 0 1
1 0 0 1 1 0 0 0
1 0 1 1 1 1 1 1
1 1 0 0 1 0 0 0
1 1 1 0 1 1 0 1

The output COLUMN and the A COLUMN are identical for all values of A, B and C; therefore
the simplification A.(B + C) + A.NOT B = A is CORRECT.

Second Example:- **Simplify A.B.NOT C + A.NOT B.NOT C**

Take out the common factor.
(A.NOT C).(B + NOT B)
Use boolean algebra x + not x = 1
(A.NOT C).1
Use boolean algebra x.1 = x
A.NOT C
CHECK USING TRUTH TABLES
C B A NOT B NOT C A.B.NOT C A.NOT B.NOT C A.NOT C OUT = A.B.NOT C + A.NOT B.NOT C
0 0 0 1 1 0 0 0 0
0 0 1 1 1 0 1 1 1
0 1 0 0 1 0 0 0 0
0 1 1 0 1 1 0 1 1
1 0 0 1 0 0 0 0 0
1 0 1 1 0 0 0 0 0
1 1 0 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0

The OUT COLUMN and the A.NOT C COLUMN are identical for all values of A, B and C; therefore
the simplification A.B.NOT C + A.NOT B.NOT C = A.NOT C is CORRECT.

Third Example:- **Simplify X.Y + Y.Z + NOT X.Z + Y.NOT Z**

Collect like terms
Y.Z + Y.NOT Z + X.Y + NOT X.Z
Take out common factor
Y.(Z + NOT Z) + X.Y + NOT X.Z
Use boolean algebra x + not x = 1
Y.1 + X.Y + NOT X.Z
Use boolean algebra x.1 = x
Y + X.Y + NOT X.Z
Take out common factor
Y.(1 + X) + NOT X.Z
Use boolean algebra 1 + x = 1
Y.1 + NOT X.Z
Use boolean algebra x.1 = x
Y + NOT X.Z
CHECK USING TRUTH TABLES
Z Y X NOT X NOT Z Y.Z X.Y Y.NOT Z NOT X.Z OUTPUT ANSWER
0 0 0 1 1 0 0 0 0 0 0
0 0 1 0 1 0 0 0 0 0 0
0 1 0 1 1 0 0 1 0 1 1
0 1 1 0 1 0 1 1 0 1 1
1 0 0 1 0 0 0 0 1 1 1
1 0 1 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 0 1 1 1
1 1 1 0 0 1 1 0 0 1 1

The OUTPUT COLUMN and the ANSWER (Y + NOT X.Z) COLUMN are identical for all values of X, Y and Z; therefore
the simplification X.Y + Y.Z + NOT X.Z + Y.NOT Z = Y + NOT X.Z is CORRECT.