Logic gates are the basic building blocks of digital computing. **A logic gate is an electrical circuit that has one or more than one input and only one output.** The input controls the output and is isomorphic with [Boolean connectives](../../Logic/Truth-functional_connectives.md) defined in terms of [truth-tables](../../Logic/Truth-tables.md).
Truth-tables present the conditions under which logical propositions are true or false. To take the `AND` operator: `AND` evaluates to `true` if both of its constituent expressions are `true` and `false` in any other circumstances (e.g. if one proposition is `true` and the other `false` (or vice versa) and if both propositions are `false` ).
The negation operator (`¬` or `~` ) switches the value of a proposition from true to false. When we put `~` before `true` it becomes false and when we put `~` before `false` it becomes `true`. We will see shortly that this corresponds to a basic on/off switch.
In this circuit, there are two transistors, each connected to a switch. The switches control the LED light. So the switches are the input and the LED is the output.
For clarity, we are not going to draw both transistors, we will simplify the diagram with a symbol for them which stands for the NAND gate:
> Remember that a 'logic gate' is a logical abstraction of a physical process: the voltage passing through a transistor. The transistors register the charge and the switches control it's flow, the 'gate' is just the combination of transistors and how they are arranged. There is not a physical gate per se, there is only the transistor whose output we characterize in terms of logic.
## Significance of the NAND gate: functional completeness
> **Equipped with just a NAND we can represent every other possible logical condition within a circuit.**
In practice, it is more efficient to use specific dedicated gates for the other Boolean connectives but in principle the same output can be achieved through NANDs alone.
The example we have looked at so far is fairly simple because there is just one NAND gate corresponding to two inputs (the two switches) and one output (the bulb).
When we add more NAND gates and combine them with each other in different ways we can create more complex output sequences and these two will have corresponding truth tables.
## `NOT` gate
This gate corresponds to the `NOT` Boolean or negation logical connective. It is really simple and derived from the trivial logical fact that `true` is `true` and `false` is `false` also known as **logical identity**.
### Natural language
>
> The negation operator (`¬` or `~` ) switches the value of a proposition from `true` to `false`. When we put `~` before `true` it becomes `false` and when we put `~` before `false` it becomes `true` .
In terms of logic gates we would create this by using a single NAND gate. Although it can take a total of two inputs, it would be controlled by a single switch, so both inputs would be set to `1 1` or `0 0` when the switch is activated and deactivated. This would remove the `AND` aspect of `NAND` and reduce it to `NOT` .
Just as we can create `NOT` logic from a NAND gate, without the `AND` conditions, we can create a circuit that exemplifies the truth conditions of `AND` without including those of `NOT`.
When we attach two NAND gates in sequence connected to two switches as input this creates the following binary conditions:
````
A B Output
_ _ _____
0 0 0 (1)
1 0 0 (2)
0 1 0 (3)
1 1 1 (4)
````
Which is identical to the truth table for `AND` :
````
p q p & q
_ _ _____
t t t (1)
t f f (2)
f t f (3)
f f f (4)
````
### Natural language
>
> `AND` (`&`) is `true` when both constituent propositions are `true` and `false` in all other circumstances viz. `false false` (`¬P & ¬Q` / `0 0` ), `true false` (`P & ¬Q` / `1 0` ), `false true` (`¬P & Q` / `0 1` )
> `OR` (in logic known as **disjunction**) in its non-exclusive form is `true` if either of its propositions are `true` or both are `true` . It is `false` otherwise.
> `XOR` stands for **exclusive or**, also known as **exclusive conjunction**. This means it can only be `true` if one of its propositions are `true` . If both are `true` this doesn't exclude one of the propositions so the overall statement has to be `false` . This is the only change in the truth conditions from `OR` .
> This is equivalent to saying 'neither' in natural language. It is only `true` both propositions are `false` . If either one of the propositions is `true` the outcome is `false` . If both are `true` it is `false`
> This one is confusing. I can see the truth conditions but don't understand them. It is `true` if both propositions are `false` like `NOR` or if both propositions are `true` and `false` otherwise.
