Boolean logic · ⁨المنطق البولياني⁩

Boolean logic 布尔逻辑 works with values that are either true or false. In electronics these are shown as 1 (true) and 0 (false). A logic gate 逻辑门 takes one or more of these inputs and gives one output, following a fixed rule.

A truth table 真值表 lists every possible set of inputs and the output for each. You build it by writing out all the input combinations.

You must know six gates. NOT has one input; all the others have two inputs (A and B).

NOT gate

The NOT gate 非门 reverses the input. Output is 1 when the input is 0.

AND gate

The AND gate 与门 gives output 1 only when both inputs are 1.

OR gate

The OR gate 或门 gives output 1 when at least one input is 1.

NAND gate

The NAND gate 与非门 is AND followed by NOT. The output is the opposite of AND.

NOR gate

The NOR gate 或非门 is OR followed by NOT. The output is the opposite of OR.

XOR gate

The XOR gate 异或门 (exclusive OR) gives output 1 when the inputs are different.

A logic expression 逻辑表达式 writes a circuit using letters and gate words. The usual way to write the gates:

For example, the expression (A AND B) OR (NOT C) means: do A AND B, do NOT C, then OR the two results together.

A logic circuit 逻辑电路 joins gates together to carry out a task. The output of one gate can become the input of another. At IGCSE a circuit has up to three inputs and one output.

You must be able to move between four forms:

• a problem statement 问题陈述 (a description in words),
• a logic expression,
• a logic circuit,
• a truth table.

From a problem statement to a circuit

Read the statement and pick out the conditions and the logic words (and, or, not). For example:

An alarm (X) sounds when the door is open (A) AND the system is switched on (B).

This is X = A AND B, so you draw one AND gate with inputs A and B.

Completing a truth table from a circuit or expression

To fill in a truth table:

1. Write all the input combinations. For three inputs there are 8 rows (000 up to 111).
2. Work out each gate's output in order, one column at a time.
3. The last column is the final output.

Adding a middle "working" column for each gate makes the final output easy to fill in. Always draw the circuit exactly as the statement says, without simplifying it.

Worked example. Complete the truth table for X = (A AND B) OR (NOT C) for the row A = 1, B = 0, C = 0. Work outwards from the brackets, one gate at a time. First A AND B = 1 AND 0 = 0, because AND needs both inputs to be 1. Next NOT C = NOT 0 = 1. Finally OR the two results: 0 OR 1 = 1. So X = 1. Give each intermediate gate its own column rather than trying to do the whole expression in one step: with three inputs there are $2^3 = 8$ rows, and those intermediate columns are where the method marks live even if the final answer slips.

• Learn all six gates and their truth tables: NOT, AND, OR, NAND (NOT AND), NOR (NOT OR), XOR (output 1 when the inputs are different).
• Build a truth table with all input rows (2 inputs → 4 rows, 3 inputs → 8), counting up in binary, and add a working column for each gate.
• Turn a problem statement into a logic expression by picking out the AND / OR / NOT words, then draw it exactly as written — do not simplify it.
• NAND and NOR give the opposite output to AND and OR; a small circle on a gate's output means the result is inverted.