## Logic Gates and Logic Circuits

Boolean logic and problem statements

Logic proposition: a statement that is either TRUE or FALSE

Problem statement: an informal definition of an outcome which is dependent on one logic proposition or a combination of two or more logic propositions

Boolean operators

Logic expression: logic propositions combined using Boolean operators, which may be written with a defined outcome

Truth tables

The truth table is a simple but powerful technique for representing any logic expression or for describing the possible outputs from a logic circuit.

A truth table is presented by making use of the convention that TRUE can be represented as 1 and FALSE can be represented as 0. The simplest use of a truth table is to represent the logic associated with a Boolean operator.

Logic circuits and logic gates

Logic gate: a component of a logic circuit that has an operation matching that of a Boolean operator

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This shows the logic gate symbols and the associated truth tables for each of the seven Boolean operators

Constructing a logic circuit from a problem statement or logic expression example

Consider the following problem statement: A bank offers a special lending rate to customers subject to certain conditions. To qualify, a customer must satisfy certain criteria:

• The customer has been with the bank for two years.

• Two of the following conditions must also apply:

• The customer is married.

• The customer is aged 25 years or older.

• The customer’s parents are customers of the bank.

To convert this statement to a logic expression you need to represent each condition by a symbol (in the same way that a problem might be tackled in normal algebra):

• Let A represent an account held for two years.

• Let B represent that the customer is married.

• Let C represent that the customer’s is age 25 years or more.

• Let D represent that the customer’s parents have an account.

The logic expression can then be written as: a AND (((BAND C) OR (BAND D)) OR (C AND D)) This could alternatively be presented with an outcome: Special rate IF AAND (((BAND C) OR (BAND D)) OR (C AN D D))

Note the use of brackets to ensure that the meaning is clear. You may think that not all of the brackets are needed.

Alternative circuits

For any given logic problem there will be different circuits that deliver the same output values from a given set of inputs. In some cases it will be possible to simplify an initial circuit design by reducing the number of logic gates.

However, reducing complexity is not just about reducing the number of logic gates. Logic circuit manufacturers can reduce costs by building circuits that contain only one type of logic gate; one that is itself cheap to manufacture. The NANO gate is an example of a universal gate which fits this requirement (the NOR gate is the other possibility). Manufacturers may find it cheaper to build a circuit with just NANO gates even though the circuit contains more components than alternatives containing different gates.

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