Consider the AND gate

• Logic is a form of human reasoning that tells us a certain proposition is true if certain preconditions are true.
• In 1854 George Boole developed a mathematical system for formulating logic statements with symbols, so the problems could be written and solved in a similar manner to ordinary algebra. His system is called Boolean Algebra and it is used in the analysis and design of digital systems.
• The basic building blocks of digital circuits are called logic gates. A gate is a circuit that performs a simple logic operation. Gates can have one, two, three or more inputs and the basic gates have a single output dependent on the inputs. Each output is also called a digital ‘bit’ of information (or ‘bit’ for short).
• The behavior of a gate can be shown in a truth table which systematically lists all the possible input states for a gate and the corresponding output states. Gates can be represented in five ways.

Consider the AND gate:

• A zero ( 0 ) corresponds to a low voltage. A one ( 1 ) corresponds to a high voltage. An inverter logic gates converts a low voltage ( 0 ) to a high voltage ( 1 ) or vice versa. Some alternative meanings for 0 and 1 are as follows:

Logic 0	Logic 1
False	True
Off	On
Low	High
Open switch	Closed switch

• Digital circuits can be put together using diodes, transistors and resistors and connected together to provide a circuit output that corresponds to the logic operations OR, AND, NOT performed on the inputs to those circuits.

• Negative logic circuits can also be used where the positive logic gate is equivalent to the complement when acting in a negative logic circuit, for example +OR is equivalent to -AND and -AND is equivalent to +OR.
• In working through the above information about logic gates, you will have noticed that each logic gate is represented uniquely. This is useful when drawing and interpreting diagrams of logic circuits.
  
  The set of symbols is as follows:

Name	Symbol	Input	Output
AND		Two signals	One signal
OR		Two signals	One signal
NOT		One signal	One signal
NAND		Two signals	One signal
NOR		Two signals	One signal
XOR		Two signals	One signal

identify that gates can be used in combination with each other to make half or full adders

• Practical circuits to perform arithmetic operations, such as addition, combine two or more gates in a circuit to provide a result.
• Two such circuits are the half-adder and full-adder. In the examples below, a bit represents the binary digits 1 or 0.

A half-adder

The need for two outputs to represent the sum of two binary 1s is obvious:

1 + 1 = 1 0

This is not ten, but two. The two digits are distinguished by their place or position relative to each other. The left-most digit is the significant bit (and is assigned the C for carry label); the right-most digit is the least significant bit (and is assigned the S for sum label).

A combination of gates in a circuit that adds two bits is called a half-adder. In the above case, this is achieved by combining an exclusive-OR and an AND gate.

A combination of gates in a circuit to add three bits is called a full-adder. The circuit below shows a combination of gates to produce a full-adder. A and B are the two inputs for this operation. CI (the third input digit) is the least significant bit from the two outputs of a separate half-adder circuit.

The term significant or least significant in front of 'bit' is necessary in order to correctly sequence the digits that represent the sum from the operation of the gates in the circuit. If all three inputs are carrying a 1, then the sum is 3 (the 1 from CI is the least significant bit from the other circuit). This is represented in binary code as 11 (see the truth table for the full-adder).

A full-adder