Boolean Algebra and Logic Gates

Boolean Algebra and Logic Gates.

Boolean Algebra. Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables. Boolean algebra traces its origins to an 1854 book by mathematician George Boole . The distinguishing factor of Boolean algebra is that it deals only with the study of binary variables. Most commonly Boolean variables are presented with the possible values of 1 ("true") or 0 ("false"). Variables can also have more complex interpretations, such as in set theory. Boolean algebra is also known as binary algebra..

Following are the important rules used in Boolean algebra. Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW. Complement of a variable is represented by an overbar (-). Thus, complement of variable B is represented as B Bar. Thus if B = 0 then B Bar = 1 and B = 1 then B Bar = 0. ORing of the variables is represented by a plus (+) sign between them. For example ORing of A, B, C is represented as A + B + C. Logical ANDing of the two or more variable is represented by writing a dot between them such as A.B.C. Sometime the dot may be omitted like ABC..

In modern Boolean algebra, we use the plus (+) symbol to mean OR, the dot (•) symbol to mean AND, and a bar above a variable to mean NOT. Note that sometimes I will use ! instead of bar to mean NOT, since it is a lot easier to make that happen in HTML, and that symbol (exclamation point or bang) is widely used to mean NOT in programming languages. Boolean variables, which can only have a value of 1 or 0, are typically capital letters like A, B, C, etc . The three basic logical operations are: AND OR NOT.

AND operation is use for logical multiplication. The dot symbol (“.”) used for representing AND operation. AND operator has an output 1 if all the inputs are 1 otherwise the output is 0..

OR operation is use for logical addition. The symbol (“+”) used for represent OR operation. OR operator has an output 1 if any of the inputs are 1 otherwise the output is 0..

NOT operation is use for complementation. The over bar symbol (“¯”) used for represent NOT operation. NOT operator has an output 1 if input is 0 or output 0 if the input is 1..

Commutative law Any binary operation which satisfies the following expression is referred to as commutative operation. ( i ) A.B=B.A (ii) A+B=B+A Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit . Associative law This law states that the order in which the logic operations are performed is irrelevant as their effect is the same. ( i ) (A.B).C=A.( B.C) ( ii) (A+B)+C=A+(B+C) Distributive law Distributive law states the following condition. A.(B+C)=A.B+A.C.

AND law These laws use the AND operation. Therefore they are called as AND laws. ( i ) A.0=0 (ii) A.1=A (iii) A.A=A (iv) A.Ā=0 OR law These laws use the OR operation. Therefore they are called as OR laws. ( i ) A+0=A (ii) A+1=1 (iii) A+A=A (iv) A+Ā=1 INVERSION law This law uses the NOT operation. The inversion law states that double inversion of a variable results in the original variable itself . Ā=A.

Basic Theorems :. 10. Boolean Theorems. OR AND NOT A+0=A A.0=0 A+Ā=1 A+1=1 A.1=A A.Ā=0 A+A=A A.A=A A+Ā=1 A.Ā=0.

Logic Gates. A logic gate is a device that acts as a building block for digital circuits. They perform basic logical functions that are fundamental to digital circuits. Most electronic devices we use today will have some form of logic gates in them. For example, logic gates can be used in technologies such as smartphones, tablets or within memory devices. In a circuit, logic gates will make decisions based on a combination of digital signals coming from its inputs. Most logic gates have two inputs and one output. Logic gates are based on Boolean algebra. At any given moment, every terminal is in one of the two binary conditions, false or true. False represents 0, and true represents 1. Depending on the type of logic gate being used and the combination of inputs, the binary output will differ. A logic gate can be thought of like a light switch, wherein one position the output is off -- 0, and in another, it is on -- 1. Logic gates are commonly used in integrated circuits (IC)..

Types of Logic Gates. AND Gate A circuit which performs an AND operation is shown in figure. It has n input (n >= 2) and one output. Y = A AND B AND C…….N Y = A.B.C……N Y = ABC….N Logic diagram Truth Table.

OR Gate A circuit which performs an OR operation is shown in figure. It has n input (n >= 2) and one output. Y = A OR B OR C…..N Y = A+B+C…….N Logic Diagram Truth Table.

NOT Gate NOT gate is also known as Inverter. It has one input A and one output Y . Y = NOT A Y = Ā Logic Diagram Truth Table.

NAND Gate A NOT-AND operation is known as NAND operation. It has n input (n >= 2) and one output . Y = A NOT AND B NOT AND C…..N Y = A NAND B NAND C……N Logic Diagram Truth Table.

NOR Gate A NOT-OR operation is known as NOR operation. It has n input (n >= 2) and one output . Y = A NOT OR B NOT OR C……N Y = A NOR B NOR C……N Logic Diagram Truth Table.

XOR Gate XOR or Ex-OR gate is a special type of gate. It can be used in the half adder, full adder and subtractor. The exclusive-OR gate is abbreviated as EX-OR gate or sometime as X-OR gate. It has n input (n >= 2) and one output . Y = A XOR B XOR C……N Y = A + B + C……N __ __ Y = AB + AB Logic Diagram Truth Table.

XNOR Gate XNOR gate is a special type of gate. It can be used in the half adder, full adder and subtractor. The exclusive-NOR gate is abbreviated as EX-NOR gate or sometime as X-NOR gate. It has n input (n >= 2) and one output . Y = A XOR B XOR C……N Y = A _ B _ C……N __ Y = AB + AB Logic Diagram Truth Table.

Logic Circuits. An electronic circuit that is consist with a number of logic gates which is use in computer t perform logical operation on its two or more input signals to produce a combined standard output signal. Logic circuit for Boolean expression [X.Y.Z].[X+Y+Z].