LearnTronics
Boolean Laws and Identities
The small set of rules that does most of the work
Boolean algebra becomes much easier when its basic identities are treated like familiar arithmetic facts.
| Name | Boolean form | Plain meaning |
|---|---|---|
| Identity | A + 0 = A A • 1 = A | OR with 0 or AND with 1 leaves A unchanged. |
| Dominance / null | A + 1 = 1 A • 0 = 0 | OR with 1 forces 1; AND with 0 forces 0. |
| Idempotent | A + A = A A • A = A | Repeating the same condition adds nothing. |
| Complement | A + ~A = 1 A • ~A = 0 | A or not-A covers all cases; A and not-A cannot both be true. |
| Double negation | ~~A = A | Invert twice and return to the original. |
| Commutative | A + B = B + A A • B = B • A | Order does not matter for AND or OR. |
| Associative | (A + B) + C = A + (B + C) (A • B) • C = A • (B • C) | Grouping does not matter when only the same operation is used. |
| Distributive | A • (B + C) = A•B + A•C A + (B•C) = (A+B)•(A+C) | Boolean algebra has two useful distributive forms. |
| Absorption | A + A•B = A A • (A+B) = A | Once A alone is enough, the more restrictive A•B adds nothing. |
Some should feel familiar
Commutative and associative laws resemble ordinary arithmetic:
A + B = B + A
A • B = B • A
and:
A • (B + C) = A•B + A•C
But Boolean algebra has its own special behavior
Because each variable can only be 0 or 1:
A + A = A
A • A = A
These are the idempotent laws. Repeating the same logical condition does not make it “more true.”
A tiny simplification
Y = A + A•B
Use absorption:
Y = A
If A by itself already turns the output on, adding the more restrictive condition “A AND B” cannot add another case.