Logic Gates simplified & deriving all gates using NAND and NOR gates

There are different types of logics. For example, we have mathematical operations that can perform certain operations on multiple inputs. These operations are known as mathematical logic. We also have logic that deals with truth, existence etc. This is known as philosophical logic. Similarly, in digital electronics, we use boolean logic. At its core, boolean logic is about classifying things as TRUE or FALSE. This means, that the output of applying boolean logic to something is one off two, true or false. This is actually perfect for digital electronics. Since digital entities deal with two values as well. 0s and 1s. Boolean logic physically manifests using logic gates.

What is Boolean logic?

Boolean logic deals with True and False. It has three main components. Also known as operators. AND, OR and NOT. These words are quite similar to their English counterparts. And these operators are the building blocks of Boolean logic. Using these three operators, we can make simple logical statements. For example, It’s NOT Monday. This implies that Monday? = 0. I would like to have bread AND cheese for breakfast. Which implies, Breakfast = 1, if bread =1 and cheese = 1.

Using these simple statements we can come up with any complex statement. After all, this is how computers and everything digital works. This is it. A basic building block of the modern Information World. Just these three operators. Next, we will understand the concept of logic gates. And then we will move on to truth tables where we will see the complete property of every logical operator starting with the basic three.

What are logic gates?

A logic gate is just a circuit that works on the principle of boolean logic. Basically, it is the physical manifestation of Boolean logic. You know how silicon (which is made from sand), is the basic building block for Integrated circuits? That is exactly how gates are made.

Transistors are electronic switches. Even their outputs are either Vcc or GND. Which can translate to 0 or 1/ True or False. We use special transistors called CMOS to make circuits that deploy the statements of a particular logic. Hence for AND logic, we have an AND gate. For OR logic, we have an OR gate and so on. We will take a look at CMOS design in our course on VLSI.

Let us now draft the truth tables for boolean logic and its corresponding logic gates. A truth table essentially shows the result when a logical operator is applied to a set of inputs. You can extrapolate these and turn them into day-to-day questions like the ones we saw in the preceding paragraph. It will help you understand the tables better which is quite necessary.

What is an AND gate?

An AND gate implements the boolean logic AND. As simple as that. The AND gate is a basic gate. Let’s take a look at the symbol and truth table for AND gate first.

and logic gate symbol

Truth table for AND operator

A B Y(A and B)
0 0 0
0 1 0
1 0 0
1 1 1

From the truth table, we can say that the output of the AND logic or an AND gate is True or high or 1, only when A and B are 1. If you observe the table, the equivalent mathematical logic for the AND boolean logic is that of multiplication. Hence, we can calculate the product of two digital inputs using an AND gate. For larger numbers, we can use the AND gate to design a circuit known as a multiplier. We will study that circuit in detail as we progress through this digital electronics course. Can you now start to see how and why logic gates are important to make computers? Moreover, we even represent the AND operation using the concept of the dot product.

Y(A and B) = A.B

Similarly, the electrical equivalent is a circuit with two resistors in series connected to a bulb. The bulb is on only when both the switches are 1 or shorted.

And boolean logic equivalent electrical circuit

What is an OR gate?

An OR gate implements the boolean logic OR. The OR gate is a basic gate. Let’s take a look at the symbol and truth table for OR gate first.

or logic gate symbol

Truth table for OR operator

A B Y(A or B)
0 0 0
0 1 1
1 0 1
1 1 1

From the truth table, we can say that the output of the OR logic or an OR gate is True or high or 1, even if either or both of A or B are 1. If you observe the table, the equivalent mathematical logic for the OR boolean logic is that of binary addition. Hence, we can calculate the sum of two digital inputs using an OR gate. For larger numbers, we use a circuit known as a full adder. We will take a look at that in the forthcoming posts. Moreover, we even represent the OR operation using the sum sign.

Y (A or B) = A + B

Similarly, the electrical equivalent of the OR gate is a circuit with two resistors in parallel connected to a bulb. The bulb is if either one or both of the switches are 1 or shorted.

OR boolean logic equivalent electrical circuit

 

What is an inverter or a NOT gate?

A NOT gate gives an output that is the inverse or opposite of its input. Hence it is alternatively known as an inverter. Think of it this way. The output of a NOT gate is not its input. Since we only have two possible outputs, it will be the opposite of the input. The inverter is one of the most important logical operators available in digital logic. Moreover, the NOT gate is the third and final basic gate. Let’s take a look at the symbol and the truth table for a NOT gate.

not logic gate symbol

Truth table for NOT operator

A Y (not A)
0 1
1 0

We can represent the not operation in the following manner

Y (not A) = A’ =  \overline { A } 

What is an EXOR gate?

An EXOR gate or EXOR logic is slightly tricky. It’s short for Exclusive OR. The easiest way I have found to remember this logic is to by remembering that an EXOR gate is also known as an “inequality detector”. Which means that the output will be high whenever the inputs are NOT equal. Let’s take a look at the symbol and the truth table for an EXOR gate.

exor logic gate symbol

Truth table for EXOR operator

A B Y(A exor B)
0 0 0
0 1 1
1 0 1
1 1 0

The EXOR gate is not a basic logic gate. We can make it using our basic gates. However, its logic is so important to the core of boolean operations that we designate it a special symbol. We can represent the EXOR operation as follows.

Y (A exor B) = A  \oplus B = AB’ + A’B

As you can see, the second equation AB’ + A’B indicates that we can implement the EXOR logic using two AND gates, two NOT gates and one OR gate. Try designing this on your own and cross check it if it’s the same as this.

What is an EXNOR gate?

exnor xnor logic gate symbol

We can extend the functionality of the gates we have seen so far by just attaching an inverter to them. This will allow us to have more options of creating complex logic using essentially the same gates that we have seen so far, albeit with an inverter attached at their outputs. Doing this, the only change in the symbols for the resulting logic gates is that we put a bubble at the output to indicate that the output shall be opposite to that of the regular output of the gate. So let’s take a look at the symbol and truth table for an EXNOR gate.

Truth table for EXNOR operator

A B Y(A exnor B)
0 0 1
0 1 0
1 0 0
1 1 1

As we can see from the truth table. the outputs of an EXNOR gate are the inverse to that of the EXOR gate. Hence, we can refer to the EXNOR gate as the “equality detector”. We can represent the EXNOR logic using the following equation

Y (A exnor B) =  \overline {A \oplus B} = A’B’ + AB

What is a NAND gate?

A NAND gate’s output is low only when both the inputs are high. In all the other cases, its output is high. We can obtain NAND logic by just connecting a NOT gate to an AND gate. Let’s take a look at the symbol and the truth table.

nand logic gate symbol

Truth table for NAND operator

A B Y(A nand B)
0 0 1
0 1 1
1 0 1
1 1 0

The NAND function is sometimes also known as the Sheffer Stroke function. We can represent it as follows

Y (A nand B) =  \overline {A.B} = A|B =  A\uparrow B

What is a NOR gate?

A NOR gate’s output is high only when both the inputs are low. In all the other cases, its output is low. We can obtain NOR logic by just connecting a NOT gate to an OR gate. Let’s take a look at the symbol and the truth table.

nor logic gate symbol

Truth table for OR operator

A B Y(A nor B)
0 0 1
0 1 0
1 0 0
1 1 0

The NOR function is sometimes also known as the Pierce function. We can represent it as follows

Y (A nor B) =  \overline {A+B} =  A\downarrow B

Why are NAND and NOR gates known as universal gates?

NAND and NOR logic gates are known as universal gates because they can implement any boolean logic without needing any other gate. They can be used to design any logic gate too. Moreover, they are widely used in ICs because they are easier and economical to fabricate.

How to design all gates using NAND and NOR logic gates?

Using universal gates we can derive all the basic logic gates, EXOR gate, and their inverse gates. It takes a bit of trial and error. Or you can use boolean logic to obtain these. You have the equation for a NAND gate and for a NOR gate. You also have equations for the gates you wish to design. Use boolean logic and solve for the output you need. It might take some time but it is necessary to practice this to get a hang of boolean logic and logic gates. Cross check your designs with the designs below. In the next post, we will design some simple combinational logic circuits using logic gates.

All gates using NAND gate
Deriving all logic gates using NAND gates

 

All gates using NOR gate
Deriving all logic gates using NOR gates

Sources:

Note: We will use all of the equations above when we code these logic gates using VHDL in our VLSI course

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