A-level Computing/WJEC (Eduqas)/Component 1/Logical operations

Logic Gates

For any decision making and calculation done by the computer on the hardware level, logic gates are used. A logic gate takes a binary input, which can either be a 1 ('switched' on/true) or a 0 ('switched' off/false). The input is then processed to give an out a binary output.

AND

A diagram of an AND gate.

The AND logic gate requires both inputs to be 1 for the output to be 1 and is represented with a point ('.'), any other combinations produce an output of 0, this is demonstrated in the truth table below.

Input		Output
A	B	A.B
0	0	0
0	1	0
1	0	0
1	1	1

OR

An OR gate.A and B are the inputs and Q is the output.

The OR logic gate requires only one input to be 1 for the output to be 1 and is represented with a plus symbol ('+'), any other combinations including a 1 will also produce an output of 1, see the truth table.

Input		Output
A	B	A+B
0	0	0
0	1	1
1	0	1
1	1	1

NOT

A traditional diagram of a NOT gate.

The NOT (Negation) logic gate, sometimes referred to as an Inverter, flips any input that it receives. For example, if a 1 is input, a 0 will be output and vice versa. It is represented using a line above the input that is being inverted.

Input	Output
A	NOT A
0	1
1	0

XOR

A XOR gate, where only one input can be 1 for the output to be 1.

The XOR (Exclusive OR) logic gate works exactly how it states in its name. Only one input can be a 1 and if any more are 1, or they are all 0, the gate will not return a 1. It is represented using a circle with a plus symbol within the circle.

Input		Output
A	B	A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

NAND

A NAND gate.

The NAND (NOT-AND) gate simply flips the logic of an AND gate. Any combination of numbers that isn't a AND, will produce a 1 whereas an AND combination will produce a 0. This is represented as A.B with a line over the two.

Input		Output
A	B	A NAND B
0	0	1
0	1	1
1	0	1
1	1	0

NOR

A NOR gate.

The NOR (NOT-OR) gate flips the logic of an OR gate. The one combination which doesn't meet the OR gate logic is a 1, where both inputs are 0. This is represented as A+B with a line over the two.

Input		Output
A	B	A NOR B
0	0	1
0	1	0
1	0	0
1	1	0

Application of Logical Operations

Masking

Masking is used to work out the state of a bit, which can either be a 0 or a 1. In the example below, the AND logical operator is used to determine the state of the 3rd most significant bit.

$10111011_{2}$AND

$00100000_{2}$Mask

___________

$00100000_{2}$

Clearing

Clearing is used to reset the contents of a register to all 0s. This can be done using the AND logical operation or the XOR logical operation. You need to know both for the exam.

AND	XOR
$10111011_{2}$AND $00000000_{2}$Mask ___________ $00000000_{2}$Cleared Register	$10111011_{2}$XOR $10111011_{2}$Mask ___________ $00000000_{2}$Cleared Register

Encryption

Encryption converts data into ciphertext. This cannot be read unless the original key used to encrypt the data is known, the process of decryption. To encrypt and decrypt data, the XOR logical operator is used on the ciphertext.

Encryption	Decryption
$10101010_{2}$XOR $11010001_{2}$Key ___________ $01111011_{2}$Ciphertext	$01111011_{2}$XOR $11010001_{2}$Key ___________ $10101010_{2}$Original Data

Simplification of Boolean Expressions