 # Boolean Algebra

### Boolean Algebra Law

The logic of boolean algebra might sound confusing but when it is broken down to bits and pieces it becomes easier to understand. The logic behind this concept is simple. You are basically dealing with 0’s and 1’s. The value of 0 is false while the value of 1 is said to be true.  In Boolean algebra, you will use only 1’s and 0’s. In the case of elementary algebra, the values of the variables are said to be numbers or alphabets and some of the operations that you can perform on them are addition and multiplication. In this article, we have simplified the concept of boolean algebra for your better understanding.

### Boolean Algebra Rules

There are basic laws that need to be followed while solving problems related to boolean algebra. The basic laws related to boolean algebra are stated below.

1. W + 0 = W

2. W + 1 = 1

3. W * 0 = 0

4. W * 1 = W

5. W + W = W

6. W + $\overline{W}$ = 1

7. W * W = W

8. W * $\overline{W}$ = 0

9. W̿ = W

10. W + (W * X) = W

11. W + ($\overline{W}$ * X) = W + X

12. (W + X) * (W + L) + W + (X * L)

### DeMorgan’s Theorem

1. $\overline{(A*B)}$ = ($\overline{(A}$ + $\overline{B}$)

2. $\overline{(A+B)}$ = ($\overline{(A}$ * $\overline{B}$)

To understand laws and theorems of boolean algebra better, some examples of the laws and theorems of boolean algebra is shown in the table below.

## Applications of Boolean Algebra

 Boolean Expression Description Equivalent Switching Circuit Boolean Algebra Law of Rule A + 1 = 1 A is in parallel with closed = CLOSED (image will be uploaded soon) Annulment A + 0 = A A is in parallel with open = OPEN (image will be uploaded soon) Identity A * 1 = A A is in series with open = A (image will be uploaded soon) Identity A * 0 = 0 A is in series with open = OPEN (image will be uploaded soon) Annulment A + A = A A is in parallel with A = A (image will be uploaded soon) Idempotent A * A = A A is in series with A = A (image will be uploaded soon) Idempotent NOT $\overline{A}$ NOT NOT A = A Double Negation A + $\overline{A}$  = 1 A is in parallel with NOT A = CLOSED (image will be uploaded soon) Complement A . $\overline{A}$  = 0 A is in series with NOT A = OPEN (image will be uploaded soon) Complement A + B = B + A A is in parallel with B = B is in parallel with A (image will be uploaded soon) Commutative A * B = B * A A is in series with B = B is in series with A (image will be uploaded soon) Commutative $\overline{A+B}$= $\overline{A}$ * $\overline{B}$ Invert and replace OR with AND DeMorgan’s Theorem $\overline{A*B}$= $\overline{A}$ + $\overline{B}$ Invert and replace AND with OR DeMorgan’s Theorem

### Solved Problems Using Boolean Algebra Laws

Question 1: Using basic rules of Boolean Algebra laws and theorems, simplify the equation ( W + X ) ( W + L )

Solution:

Q = ( W + X ) ( W + L )

Q = ( W * W ) + ( W * L ) + ( W * X ) +(  X * L ) . . . . . . . . . Distributive Law

Q = W + (  W * L ) + (  W * X ) + (  X * L ) . . . . . . . . . . . . . Idempotent AND law ( W * W = W )

Q = W ( 1 + L ) + ( W * X ) + ( X * L ) . . . . . . . . . . . . . . . . Distributive Law

Q = ( W * 1 ) + ( W * X ) + ( X * L ) . . . . . . . . . . . . . . . . . . Identity OR Law ( 1 + L = 1 )

Q = W ( 1 + X ) + ( X * L ) . . . . . . . . . . . . . . . . . . . . . . . . Distributive Law

Q = ( W * 1 ) + ( X * L ) . . . . . . . . . . . . . . . . . . . . . . . . . . Identity OR Law ( 1 + X = 1 )

Q = W + ( X * L ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identity AND Law ( W * 1 = W )

1) What are Boolean Algebra Rules and Theorems?

There are basic laws that need to be followed while solving problems related to boolean rules for simplification. The basic laws related to boolean algebra are stated below.

• W + 0 = W

• W + 1 = 1

• W * 0 = 0

• W * 1 = W

• W + W = W

• W + W̅ = 1

• W * W = W

• W * W̅ = 0

• W̿ = W

• W + ( W * X ) = W

• W + ( W̅* X ) = W + X

• ( W + X ) * ( W + L ) + W + ( X * L )