**Logic Equations**

In addition to portraying the operations of a logic gate through truth tables and grammatical definitions, logic equations serve as a means to represent not only logic gates and circuits but also, employ theorems and equivalences, to streamline equations by reducing the number of terms. In the realm of logic equations, each Boolean variable is assigned a letter or symbol, akin to the algebraic representation of unknown numerical values using letters—a method referred to as Boolean algebra.

Symbolic logic encompasses values, variables, and operations;

TRUE is denoted as 1,

FALSE as 0.

Variables are represented by letters and can assume values of 0 or 1. Operations function with one or more variables.

**AND Gate Equation**

The AND gate operation can be expressed through a Boolean algebraic equation. For a 2-input AND gate, the equation is:

[ X = A .B ]

The symbol for the AND gate operation is a centre dot, not implying multiplication. The equation signifies that the output (X) is logic 1 when both A and B inputs are in their 1 states.

** **

**OR Gate Equation**

The Boolean algebraic expression for an OR gate is given by:

[ X = A + B ]

This equation indicates that the output (X) is logic 1 when either A or B inputs, or both, are in their 1 states.

**NOT Gate Equation**

The NOT gate operation is represented by the Boolean algebraic equation:

[ X = A}

A complement bar is placed over the assigned input letter, denoting that the output state (X) is the opposite of the logic state applied to the input.

**Uses Of Logic Gates**

Logic gates serve as the foundational components of digital electronics, created through the combination of transistors to perform various digital operations (e.g., Logical OR, AND, NOT). Virtually every digital device, including computers, mobile phones, calculators, and digital watches, incorporates logic gates. To illustrate, consider the single-bit full adder in digital electronics—a logic circuit that executes the logical addition of two single-bit binary numbers.

These are gates that are formed from combination of two logic gates. There are two types of alternative logic gate:

## NAND GATE

A NAND gate is the combination of an AND gate and NOT gate. It operates the same as an AND gate but the output will be opposite. Remember, the NOT gate does not always have to be the output leg; it could be used to invert an input signal also.

**LOGIC SYMBOL FOR THE “NAND” GATE**

NAND Gate

Notice the circle on output C.

TRUTH TABLE FOR THE “NAND” GATE

INPUT INPUT OUTPUT

A B C

0 0 1

0 1 1

1 0 1

1 1 0

**NAND GATE EQUATION**

The NAND gate operation can also be expressed by a Boolean algebra equation. For a 2 – input NAND gate, the equation is:

X = A.B

This equation read X equal to A and B NOT, which simply means that the output of the gate is not a logic 1 when A and B inputs are their 1 states.

**NOR GATE**

A NOR gate is the combination of both an OR gate and NOT gate. It operates the same as an OR gate, but the output will be the opposite.

NOR Gate

TRUTH TABLE FOR THE “NOR” GATE

INPUT INPUT OUTPUT

A B C

0 0 1

0 1 0

1 0 0

1 1 0

NOR GATE EQUATION

The NOR gate operation can also be expressed by a Boolean algebra equation. For a 2 – input NAND gate, the equation is:

X = A + B

The expression is the same as the OR gate with an over bar above the entire portion of the equation representing the input. This equation read X equal to A or B NOT, which simply means that the output of the gate is not a logic 1 when A or B are in their 1 states.

**Uses Of Logic Gates**

Logic gates are in fact the building block of digital electronics, they are formed by the combination of transistors (either BJT or MOSFET) to realise some digital operations like logical OR, NOT, AND etc. Every digital product like computers, mobile phones, calculators, even digital watches contains logical gates.

XOR GATE

The XOR (exclusive – OR) gate acts in the same way as the logical “either or”. The output is “True” if either but not both, of the inputs are “true”. The output is “false” or if both inputs are “true”.

LOGIC SYMBOL FOR “XOR” GATE

XOR Gate

TRUTH TABLE FOR THE “XOR” GATE

INPUT INPUT OUTPUT

A B Y

0 0 0

0 1 1

1 0 1

1 1 0

**XOR Comparator**

Comparator is a combinational logic circuit that compares the magnitudes of two binary quantities to determine which one has the greater magnitude. In order word, comparator determines the relationship of two binary quantities. A XOR can be used as basic comparator.

As you can see, the only difference between these two symbols is that the XNOR has a circle on its output to indicate that the output is inverted.

**XOR Combination**

One of the most common uses for XOR gates is to add two binary numbers. For this operation to work, the XOR gate must be used in combination with an AND gate.

XNOR Combination1

To understand how the circuit works, review how binary addition works:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10

If you wanted, you could write the results of

each of the preceding addition statements by

using two binary digits, like this:

0 + 0 = 00

0 + 1 = 01

1 + 0 = 01

1 + 1 = 10

When results are written with two binary digits, as in this example, you can easily see how to use an XOR and an AND circuit in combination to perform binary addition.

If you consider just the first binary digit of each result, you’ll notice that it looks just like the truth table for an AND circuit and that the second digit of each result looks just like the truth table for an XOR gate.

The adder circuit has two outputs. The first is called the Sum, and the second is called the Carry. The Carry output is important when several adders are used together to add binary numbers that are longer than 1 bit.

Related Posts:

Differences Between Primary and Secondary Memory