What is the difference between boolean logic and binary arithmetic

Logic gates may have two or more inputs and, except in some special cases, they have a single output. The status of the input and output terminals can only be in one of the two binary conditions, either low 0 or high 1 , represented by two different voltage levels, typically 0 volts for logic 0, and around 3 to 5 volts for logic 1, depending on the semiconductor technology used.

Logic gates also require a power supply. Electronic gates are generally constructed from transistor circuits which depend for their operation on the use of the transistor as a switch, rather than as an amplifier for which it was originally invented.

With no voltage on the base, there is no current through the transistor which is thus switched off and the output collector voltage will be high.

When a "high" voltage is applied to the base the transistor is switched on and output collector voltage will be "low". See more on the page about semiconductors.

An early version of a bi-stable switching circuit was the Eccles and Jordan flip-flop based on valves vacuum tubes. The later transistor version was one of the first electronic circuits to be implemented as an Integrated Circuit by Robert Noyce in Flip-flops rely on the concept of feedback in which the output of a circuit is fed back into the input such that when the input is high, the output is low and vice versa. See below an example of transistor switches used in the electronic circuit used to implement a three input NOR gate.

Logic circuit truth tables show the status of the output terminal or terminals of logic gates and logic circuits for all the possible input combinations. The gate or circuit's input states are shown in the left columns of the table while the corresponding output states are shown in the right columns. The Vernam cipher is a special application of XOR logic. Also called Modulo 2 Addition, it is similar to a digital adder except that the carry digits are ignored.

An important cryptography tool, its special property is that a plaintext message string can be enciphered by XORing it with a random symbol scrambler string or key of the same length to create truly unbreakable ciphertext. The ciphertext can however be deciphered directly by XORing it with the original scrambler key.

In the example opposite:. The diagram opposite is an example of a three input NOR gate showing the electronic circuit from which the gate is constructed together with the circuit symbol for the gate and the truth table associated with the gate. Applying a logic 1 to any of the input terminals A, B or C cause a current to flow through the load resistor R4 which in turn causes the voltage on the output terminal Z to fall to logic level 0.

Only when all the input terminals are set to logic 0 will the current through the load resistor be cut off and the voltage on the output terminal will rise to logic level 1.

Boolean logic is used to design complex digital circuits to perform a wide variety of logical functions. There is however often more than one way to implement a logic circuit by using alternative types of gates.

Some examples follow. The Set-Reset flip-flop constructed from two cross connected, two input, NOR gates is one of the fundamental digital logic circuits. It is a bi-stable circuit which can store a single data bit in the form of a binary zero or a binary one and is used as a memory device or a latch.

Applying a logic1 to the Set terminal S stores a 1 and sets the output terminal Q to logic 1. Applying a logic 1 to the Reset terminal R clears the memory, storing a 0 instead and sets Q to logic 0. Registers are common storage devices providing temporary storage of multi-bit data words such as 4, 8 or 16 bit words.

They are constructed from groups of flip-flops each storing a single bit of information so that n flip-flops are used to store an n bit word. Half adders have only two inputs for the two bits to be added and can not accept a carry bit from a previous stage. They do however have two outputs, one for the Sum and the other for the Carry output from the two bit addition. The half adder opposite is another example of how logic functions can be implemented in different ways.

In this case the adder circuit can be simplified by using only two gates, an AND gate and an XOR gate to perform the same half adder function as the circuit above.

Full adders are designed to accept a carry bit from a previous stage and hence have three inputs. The circuit below is a an example of single bit, full adder constructed entirely from two input NOR gates. In this case it is essentially two, two-input, half adders in series with the input carry bit bypassing the first adder and being added to the sum of the two input bits from the first adder, in the second adder. Note that it takes 12 such gates simply to add two single bits plus any input carry bit from a previous addition stage and to provide two output bits representing the sum of the bits and any associated carry bit.

A logic circuit designed to add two eight bit words will require eight times as many gates. It may seem strange to use so many gates when the circuit could easily be implemented with fewer, more complex gates, but circuits such as the adder above were used in the Apollo Guidance Computer which took the US astronauts to the Moon in All of its digital circuits were built from three input NOR gates.

This was because they needed highly reliable semiconductor components and at the time when the computer design was frozen, integrated circuit technology was still in its infancy and NASA wanted to limit the number of different components used to those which had a proven track record. NOR gates were chosen because they were one of the very few options that met this requirement and because NOR gates were simple and more versatile than other available gates for building more complex functions.

A binary adder can be adapted to perform other arithmetic operations such a subtract, multiply and divide as well as other more complex mathematical functions, avoiding the need for multiple specialist processors, by making use of the following principles of binary arithmetic. This operation involves m - n steps where m is the number of bits in the dividend the number being divided and n is the number of bits in the divisor.

Note: Checks must be included and the operation set up to avoid the potential problem of dividing by 0. Key Difference: In the field of computers and electronics, Boolean refers to a data type that has two possible values representing true and false. Binary in mathematics and computers, refers to a base 2 numerical notation. Thus, both are based on the same underlying concept but used in context to different systems. Comparison between Boolean and Binary: Boolean Binary Definition In the field of computers and electronics, Boolean refers to a data type that has two possible values representing true and false.

NOT x returns True if x is false or null and False if x is true. If x and y are both true or false, the statement will return False. A binary number system is also called as base-2 number system. Binary multiplication uses the same technique as decimal multiplication. To perform binary division follow the following steps: Step 1 - Align the divisor Y with the most significant end of the dividend.

Step 3 - Compare X and Y. Step 4 - Shift Y one bit to the right and go to step 2. In this binomial options pricing model , where there are only two possible outcomes, the Boolean variable represents an increase or a decrease in the price of the security.

This type of modeling is necessary because, in American options, which can be exercised at any time, the path of a security's price is just as important as its final price. The binomial options pricing model requires the path of a security's price to be broken into a series of discrete time ranges. As such, the binomial options pricing model allows an investor or trader to view the change in the asset price from one period to the next.

This allows them to evaluate the option based on decisions made at different points. Because a U. An analysis of the binomial tree would allow a trader to see in advance if an option should be exercised. If there is a positive value, then the option should be exercised, if the value is negative, then the trader should hold onto the position.

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