Computing - Logic Gates
AQA Computer Science 2022
Logic Gates
| Name | Shape |
|---|---|
| AND | Like a “D” |
| OR | Like a crescent |
| NOT | Like a triangle with a dot |
| NAND | Like and with a dot |
| NOR | Like an OR with a dot |
| XOR | Like OR with an extra crescent |
AND
| Input 1 | Input 2 | Output |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR
| Input 1 | Input 2 | Output |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
NOT
| Input 1 | Output |
|---|---|
| 0 | 1 |
| 1 | 0 |
NAND
AND + NOT
| Input 1 | Input 2 | Output |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
NOR
OR + NOT
| Input 1 | Input 2 | Output |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
XOR
OR, but 1 & 1 is 0.
XOR is also sometimes called the NEQ operator, meaning “not equivalent”. This is because you can also describe it as only being true if the operands differ.
| Input 1 | Input 2 | Output |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
XOR is probably the most popular logical operator in exams. The property that makes this useful is that XOR is like adding two binary numbers together in a single column:
| 0| | 1| | 1|
| 0| | 0| | 1|
|--| |--| |--|
| 0| | 1| | 0| (1 carried over)
XOR as an adder
In terms of the operands, how could you describe XOR?
XOR is true if the operands are different.
Why is XOR sometimes called the NEQ operator?
Because it only is true if the operands are different.
What property makes XOR useful?
It’s like adding two binary digits together in a single column.
Half-Adders
A one-bit binary adder. This means it can deal with adding two one-bit binary numbers together and dealing with carries.
They take two inputs:
- A, the first binary digit.
- B, the second binary digit.
And have two outputs:
- S, the sum of the two binary digits.
- C, the carry from the addition if there was one.
| A | B | C | S |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 |
What outputs does a half adder have?
- _ _ S _ _ , the sum of the binary digits.
- _ _ C _ _ , the carry from the sum if there was one.
What is the limitation of a half adder?
Although it outputs a carry, it cannot take a carry as input.
What is a half adder that can take a carry as an input called?
A full adder.
Full Adders
Full adders take three inputs:
- A, one bit
- B, one bit
- C in, a carry bit
They have two outputs:
- Sum, the result of adding the two bits and the carry (think “left hand column”)
- C out, the carry.
| A | B | C in | C out | Sum |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 |
Notice how the truth table is just like adding three numbers together. You can think of a full adder as a half adder that then has to deal with a carry in bit too. There’s this really beautiful way that a full adder is actually just two half adders and an OR gate and it’s in my head but I can’t get it out.
How can you conceptualize a full adder in the context of a half adder?
A full adder is like a half adder that then has to deal with a carry in bit.
What are the inputs to a full adder?
- _ _ A _ _ , a binary digit
- _ _ B _ _ , a binary digit
- _ _ C in _ _ , the carry in
What outputs does a full adder have?
- _ _ S _ _ , the sum
- _ _ C out _ _ , the carry from the addition if there was one.
Why is it important to have a carry in and a carry out in a full adder?
Because adders for multiple bits work as a chain, where the previous carry out is passed on as the carry in.
How can you create a multiple bit adder?
- Visualise the multiple bits as two columns, one for each number.
- The first column only needs to be a half-adder because there’s no carry in.
- The rest of the columns are full adders, taking the two bits from that column and the previous carry out.
In what way is a full adder two half-adders and an OR gate?
- Inputs A and B are run through a half adder, if this overflows then a carry is needed
- The sum of A and B and the carry in are run through a half adder, if this overflows then a carry is needed