Logic gates
| English | Chinese | Pinyin |
|---|---|---|
| logic gate | 逻辑门 | luó jí mén |
| truth table | 真值表 | zhēn zhí biǎo |
| Boolean | 布尔 | bù ěr |
| logic expression | 逻辑表达式 | luó jí biǎo dá shì |
The master's thesis that built the digital world
- In 1937 a 21-year-old student, Claude Shannon, noticed that the on-off relays in telephone exchanges behaved exactly like the true-false algebra George Boole had written a century earlier.
- His thesis showed that any logical statement could be built as a circuit of switches, and any circuit of switches described as a logical statement.
- Every processor made since is a very large number of those switches, arranged into a handful of standard logic gates 逻辑门.
- This lesson is the handful: six gates, their symbols, and their truth tables.
What a logic gate is
- A logic gate is a small circuit that carries out one Boolean 布尔 operation. Every input and every output is either 0 (false, low voltage) or 1 (true, high voltage).
- For each gate you must know three things: its symbol, its function in words, and its truth table 真值表, which lists the output for every combination of inputs.
- Every gate in this course has two inputs except NOT, which has one.
The six gates: a shape for each, and a bubble where the output is inverted
Logic gates
output from the truth table
Toggle A and B through a gate and watch the truth table row light up.
NOT, AND, OR
- NOT inverts its single input: 0 becomes 1 and 1 becomes 0.
- AND outputs 1 only when both inputs are 1.
- OR outputs 1 when at least one input is 1, so it is 0 only when both inputs are 0.
| A | B | A AND B | A OR B |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
A NOT gate outputs 1 when its input is 0.
NOT inverts: 0 → 1 and 1 → 0.
An AND gate outputs 1 when:
AND outputs 1 only if every input is 1; otherwise 0.
An OR gate outputs 0 only when:
OR is 1 if at least one input is 1, so it is 0 only when every input is 0.
NAND, NOR, XOR
- NAND is NOT AND: the output is 0 only when both inputs are 1, and 1 otherwise.
- NOR is NOT OR: the output is 1 only when both inputs are 0.
- XOR (exclusive OR, also written EOR) outputs 1 when the inputs are different, and 0 when they are the same.
| A | B | A NAND B | A NOR B | A XOR B |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 0 | 0 | 0 |
A XOR gate outputs 1 when:
XOR (exclusive OR) is 1 when the inputs differ (0,1 or 1,0) and 0 when they are the same.
A NAND gate outputs 0 only when:
NAND = NOT AND, so it is 0 exactly when AND would be 1 (all inputs 1), and 1 otherwise.
A NOR gate with both inputs 0 outputs what value (0 or 1)?
NOR = NOT OR. OR of (0,0) is 0, so NOR is 1. NOR is 1 only when all inputs are 0.
Worked example: building a truth table
- Task: construct the truth table for a NOR gate.
- Write the input columns A and B and list every combination in binary counting order: 00, 01, 10, 11. Four rows, never three, never repeated.
- Apply the rule to each row. NOR is 1 only when both inputs are 0, so the output column reads 1, 0, 0, 0.
- The order matters for the marks: the examiner compares your output column with the scheme row by row.
Put the rows of a two-input truth table in the order the examiner expects.
Binary counting order: 00, 01, 10, 11. The output column is then compared with the mark scheme row by row.
Worked example: defining a gate in words
- Task: define the function of an XOR gate.
- The mark-scheme wording: the output is 1 when the inputs are different, or equivalently when exactly one input is 1; the output is 0 when both inputs are the same.
- For NAND: the output is 0 only when both inputs are 1. For NOR: the output is 1 only when both inputs are 0.
- "Only when" and "at least one" are doing the work in these sentences. Leave them out and the definition also describes another gate.
Reading a symbol
- AND has a flat back and a rounded front, like a D. OR has a curved back and a pointed front. XOR is OR with an extra curved line across its inputs.
- A small circle, the bubble, on the output means "invert". AND with a bubble is NAND, OR with a bubble is NOR.
- NOT is a triangle with a bubble: one input, one output.
Flat back for AND, curved back for OR, an extra line for XOR, a bubble for NOT
Match each description of a symbol to its gate.
The back of the shape says AND or OR; the extra line says exclusive; the bubble says inverted.
Recognising a gate from its output column
- Read the output column from the 00 row down to the 11 row.
- 0 0 0 1 is AND. 0 1 1 1 is OR. 0 1 1 0 is XOR.
- 1 1 1 0 is NAND. 1 0 0 0 is NOR. The inverted gates begin with a 1 where their partner begins with a 0.
For inputs A = 0 and B = 1, which gates output 1? Select all that apply.
One input is 1, so OR and XOR give 1; both are not 1, so AND gives 0 and NAND gives 1; both are not 0, so NOR gives 0.
Writing the logic expression
- A logic expression 逻辑表达式 writes the gate as words:
X = A AND B,X = A OR B,X = NOT A,X = A NAND B,X = A NOR B,X = A XOR B. - Brackets show which operation happens first when gates are joined:
X = (A AND B) OR C. - The expression, the truth table and the circuit are three views of one function. Next lesson joins the gates into circuits and moves between all three.
A circuit outputs 1 only when both of its inputs are 1. Which logic expression describes it?
"Only when both" is the AND definition. NAND is its inverse, OR needs only one input, XOR needs them to differ.
Marks that slip away
- XOR is not OR. For inputs 1 and 1, OR gives 1 and XOR gives 0.
- NAND is NOT (A AND B), not (NOT A) AND B. The bubble is on the output.
- Truth-table rows go 00, 01, 10, 11. A table in another order is marked as wrong even when every row is right.
- A NOT gate has one input. Every other gate in this course has exactly two.
You've got it
- a gate does one Boolean operation on inputs of 0 and 1; know each symbol, function and truth table
- AND = both 1 · OR = at least one 1 · NOT = invert · NAND = 0 only when both 1 · NOR = 1 only when both 0 · XOR = inputs differ
- the bubble on a symbol inverts the output; truth tables run 00, 01, 10, 11
- a logic expression writes the same function in words:
X = A NAND B