# Boolean Simplification Examples

[latexpage]

## 1

### Example 1

Simplify the following Boolean Algebra:

$(A.\overline{A} )+B$

hint
• Complement Rule
• Identity Rule
Solution

$(A.\overline{A} )+B$

$(0)+B$#Complement

$B$ #Identity

## 2

### Example 2

$(A.B)+(\overline{A}.B)$

hint

Absorption

Solution

$(A.B)+(\overline{A}.B)$ #Absorption

B

## 3

### Example 3

$(A+B).(A+C)$

hint
Solution

## 4

### Example 4

$(\overline{A} + B).(A+B)$

hint
Solution

## 5

### Example 5

$A\overline{C} + ABC$

hint
Solution

## 6

### Example 6

$A.\overline{B}.D + A.\overline{B}.\overline{D}$

hint
Solution

## 7

### Example 7

$B + B.C + A.C + A.B + A.C.B$

hint
Solution

## 8

### Example 8

$\overline{A}.\overline{B}.\overline{C}.\overline{D} + \overline{A}.\overline{B}.\overline{C}.D + \overline{A}.\overline{B}.C.D+\overline{A}.\overline{B}.C.\overline{D} + \overline{A}.B.\overline{C}.\overline{D}$

hint
Solution

## 9

### Example 9

$\overline{(\overline{A}.(B+C))}$

hint
• Use De Morgan’s Law
• Use Involution Law(Double Inversion)
Solution

$\overline{(\overline{A}.(B+C))}$ #Original Expression

$(\overline{\overline{A}} + \overline{(B + C))}$ #Use De Morgan’s law on the ANDed components

$(A + \overline{(B + C))}$  Involution Law (Double negatives cancel)

$(A + (\overline{B}.\overline{C}))$ De Morgan’s Law again

$\bf{A + \overline{B}.\overline{C}}$  # BODMAS rules allow removal of brackets

hint
Solution

hint
Solution

[latexpage]