## 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.