What are various Boolean identities?

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What are various Boolean identities? Draw the ‘AND’ circuit using diode logic.

How do you prove Boolean identities?, What are the different Boolean algebra rules?, What is De Morgans theorem?, How do you simplify Boolean expression?

Laws of Boolean algebra are used for simplifying a complex Boolean expression. The different Boolean laws are discussed as follows :
1. OR law:

The following theorems implement the OR law :
(i) Out of two variables to be ORed, if anyone variable is low (0), then the output value is the
other variable i.e.

A + 0 = A

• Out of two variables to be ORed, if anyone variable is high (1), then the output value is high i.e. A + 1 = 1
• If both variables to be ORed have the same value, then the output value is of one input variable i.e.

A + A = A

• Out of two variables to be ORed, if anyone variable is the complement of the other, then the output value is high i.e.

A + A(bar) = 1

2. AND law: The following theorems implement the AND law :

 Out of two variables to be ANDded, if anyone variable is low (0), then the output value is zero i.e.

A × 0 = 0

• Out of two variables to be ANDed, if anyone variable is high (1), then the output value is other variable i.e.

A × 1 = A

• If both variables to be ANDed have the same value, then the output value is also the same as one of the variable i.e.

A × A = A

• Out of two variables to be ANDed, if anyone variable is the complement of the other, then the output value is zero i.e.

A × A(bar) = 0

3. Commutative law:

It states that changing the sequence of the variables to perform logic operation does not change the output, which means that change of variable sequence is allowed i.e.

A + B = B + A

A × B = B × A

4. Associative law: It states that changing the order of logic operations does not change the output, which means that changing the order of logic operations is allowed i.e.

 A + ( B + C ) = ( A + B ) + C

A × ( B × C ) = ( A × B ) ×C

 

5. Distributive law: It states that ANDing the result of several ORed variables with a single variable is the same as ANDing result with an individual variable with each of the multiple variables, and its product is an ORed variable i.e.

 A × ( B + C ) = A × B + A × C

A + ( B × C ) = ( A + B ) × ( A + C)

A + ( A(bar) × B ) = ( A(bar) + A) × ( A + B ) = 1 × ( A + B ) = A + B

A + ( A(bar) × B ) = A + B

5. Complementation law: It states that the complement of 0 is 1, of 1 is 0, and of A is A(bar) i.e.

 (bar) =1

 1 (bar)= 0

A(bar) = A

A × A(bar) = 0

A + A(bar) = 1

7. Absorption law: It states the following properties :

A + AB = A

A( A + B ) = A

8. Idempotency law: It states the following properties :

 A + A = A A × A = A

 9. Inversion law: It states that if a variable is subjected to a double inversion than it will result in the original variable itself i.e.

A (bar)(bar)= A

AND gate circuit using diode :

In Fig.1 for positive logic, if any of the three inputs are at 0 V (logic ‐ 0), the corresponding diode becomes forward biased or conducting showing zero resistance and hence the voltage at Y becomes zero. If all the inputs A, B, and C are at +5 V (logic‐1), the diodes are in reversed bias, hence no diode conducts and the voltage at Y will be +V (logic‐1) This is described with the help of the truth table shown in Fig.2