## 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

**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

**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*

**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*

**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*

**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

**Absorption law:**It states the following properties :

*A *+* AB *=* A*

*A*(* A *+* B *) =* A*

**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

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