Operator precedence

Also called: precedence, order of operations, Boolean precedence, NOT AND OR order

The rule for which Boolean operation is done first when there are no brackets: NOT first, then AND, then OR.

When an expression mixes operators, you need to know which one binds tightest. Boolean algebra uses the same idea as arithmetic, where × comes before +:

NOT (the bar or prime) first,
then AND (letters side by side, or ·),
then OR (+).

So means A OR (B AND (NOT C)). Brackets override the order: does the OR first.

A long bar acts like brackets. In you work out first and then flip the result. A short bar covers only the one letter under it, so flips only B.

Why it matters: the same letters grouped differently give different functions. Many errors, especially when typing answers or applying de morgans laws, come from regrouping by accident. If you are unsure, add brackets: they never hurt.


0	0	0	0	0
0	0	1	0	0
0	1	0	0	0
0	1	1	1	1
1	0	0	1	0
1	0	1	1	1
1	1	0	1	0
1	1	1	1	1

Worked examples

Example

Grouping and evaluating [[A + BC']]

Evaluate at A = 0, B = 1, C = 0.

1.
NOT: C = 0, so = 1.
2.
AND: = 1 · 1 = 1.
3.
OR: 0 + 1 = 1.
4.
Result: 1.

Example

Same letters, different grouping

Compare with at A = 1, B = 0, C = 0.

1.
: AND first gives = 0, then 1 + 0 = 1.
2.
: the bracket first gives 1 + 0 = 1, then 1 · 0 = 0.
3.
The outputs differ, so the two expressions are different functions. The truth table above shows they disagree in rows 100 and 110.

Common mistakes

Reading left to right. is not (A + B)C; AND comes before OR.
Stretching a short bar. In the bar covers only B, so it is not .
Dropping brackets after De Morgan. becomes , but needs , with the brackets kept.