In theory, there is no limit to the number of gates that can be arrayed together in a single device. But in practice, there is a limit to the number of gates that can be packed into a given physical space.

His approach has many similarities to ordinary algebra, and has found application in the design of computer circuitry, a topic I will discuss below. In Boolean algebra, statements are represented by letters, just as unknown numbers are in ordinary algebra.

Statements can be combined with other statements to make compound statements which themselves are either true or false. We will assign the value 1 to any true statement and 0 to any false statement.

Then the truth or falsity of a compound statement can be derived from mathematical operations very much like, but not identical to, the familiar mathematical operations of addition and multiplication. The two most basic compound statements between two statements a and b are the AND relationship and the OR relationship.

To the right is the truth table for a AND b. We will represent an AND statement as multiplication: If either a or b or both are true, then a OR b will equal 1. Here is the truth table for a OR b. We represent a OR b as addition. If both a and b are true, the statement a OR b is still just simply true.

We can now write down some useful identities for these operators: Both of these operations are associative in the usual algebraic sense: In either case, the parentheses are unnecessary.

Similarly, both operations are commutative in that the statements can be taken in either order: Recall that ordinary arithmetic has a distributive rule for combined addition and multiplication.

There are two such rules in Boolean algebra: The left side would be 15 while the right side would be 8. The inversion operation can be moved into and out of parentheses using the following important identities: Another very useful expression involving two statements a and b is the following: It is 1 when either a or b is 1, but not when both are 1.

If both a and b are true, the parenthesis on the right is 1 and this becomes 0 when inverted. Before moving on to look at computer circuits, there is one more simplification to work through.

Generally, in a computer circuit, 1 and 0 values are represented by a higher and a lower voltage on a wire.

An AND Gate, for example, accepts voltage inputs on two wires, and generates a voltage on an output wire, based on the AND truth table. Recall that numbers in a computer are represented in binary form, with the digits 1 and 0 strung out to represent various power of 2.

The circuit will have to work on a single column of binary digits at a time. So our circuit must do two things: There are two inputs and two outputs to consider. Let a be the first input bit either 0 or 1 and b be the second input.

Let d1 be the digit output and c1 be the carry digit. Then we can describe the necessary logic relations as follows: It accomplishes what we asked it to do, but that would only work for the first column of the sum. Then the final carry is just c1 OR c2. We can express this by three more equations: The three terms pick up the three possible ways in which two of the three inputs are 1.

Notice that the final expression for the carry is not as simple as it might be. The inverted terms are not really necessary as they contribute nothing. This rather long and probably tedious example illustrates how Boolean algebra is related to logic circuitry in computers. It helps to analyze the outputs from any logic circuit, and can actually be used to simplify the number and type of gates needed to implement a particular logic function.Any Boolean algebra operation can be associated with an electronic circuit in which the inputs and outputs represent the statements of Boolean algebra.

Although these circuits may be complex, they may all be constructed from three basic devices. Scribe: Addition and Boolean Algebra Essay Boolean algebra and circuit optimization Scribe for lecture on by Greeshma Balabhadra Sai Sumana P – Priya - INTRODUCTION Boolean algebra provides the operations and the rules for working with the set {0,1}.

Boolean algebra (named after the mathematician George Boole) is a form of aritmetic that deals solely in ones and zeroes. It has only three operators: addition, multiplication and negation. As we shall see these correspond to OR, AND and NOT repectively. XOR is sometimes included for convenience.

scribe timing and logical behavior of a circuit simultaneously, which is a basic research topic of by taking real values with operations of addition, subtraction and multiplication.

1 Boolean process Boolean algebra can be interpreted in terms of sets or logical propositions as well as switching. Boolean algebra and circuit optimization Scribe for lecture on by Greeshma Balabhadra Sai Sumana P Priya - In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 benjaminpohle.comd of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra .

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Boolean Algebra | Math Junky