Basic operations such as And, Or and Not
Logic operations are used on binary data to calculate logic information about their values. Each operation is represented by a truth table which gives at least one output based on the inputs it receives. There are three basic operations (And, Or and Not) from which all other operations can be built, but are expressed separately for convenience.
The and operation takes in two values and gives a true (1) value if both inputs are true, otherwise it is false (0). And is represented by the ˄ symbol (NB: the symbol is an Up Arrowhead, not a hat).
The function can be defined as:
| Value 1 | Value 2 | And |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
And is a commutative operation.
The or operation takes in two value and give a true value if at least one of the input values is true. It is represented by ˅.
The function can be defined as:
| Value 1 | Value 2 | Or |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Or is a commutative operation.
The not operator takes a single input value and inverts the value of it. It is represented by the ¬ symbol, but is sometimes also represented by a ! before the value or an overline.
| Value | Not |
|---|---|
| 0 | 1 |
| 1 | 0 |
The nand operation combines the And and Not operations, so provides the opposite result to an and operation.
| Value 1 | Value 2 | Nand |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Nand is a commutative operation.
The nor operation combines Or and Not to invert the result of an or operation.
| Value 1 | Value 2 | Nor |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
Nor is a commutative operation.
The xor operator will only give true if a single input is true, and for all other cases (neither true or both true) it is false. Xor can be constructed from the equation (A ˅ B) ˄ ¬(A ˄ B). Xor either represented as “xor” or the ⊕ symbol.
| Value 1 | Value 2 | Xor |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Xor is a commutative operation.
The implication operator is true if the first input is sufficient for the second. Whilst using in some logic expressions this is less likely to be implemented in a language. Implication can be constructed from the equation ¬A ˅ B, and is represented by the symbol ⇒.
| Value 1 | Value 2 | Implies |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Double implication is where each input implies the other. This results in the same behaviour to an equality operator, but can be more correctly expressed as (A ⇒ B) ˄ (B ⇒ A). It is expressed by the symbol ⇔. It is also referred to as if and only if, or iff.
| Value 1 | Value 2 | Double Implies |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Double implication is a commutative operation.