Boolean Logic · 5 question types
Past paper frequency (2018 to 2024)
This topic accounts for approximately 6% of your exam marks.
Writing Boolean expressions from logic diagrams and simplifying using laws appear regularly.
Logic circuits and Boolean expressions can be rearranged using a small set of laws. Knowing them lets you turn a complicated expression into a simpler one with the same truth table, which often means a smaller, cheaper circuit.
The laws below come in pairs; each pair has one law for AND (·) and one for OR (+).
A · 1 = AandA + 0 = A
ANDing with 1 leaves the value unchanged; ORing with 0 leaves the value unchanged. These are the Boolean analogues of x × 1 = x and x + 0 = x in ordinary algebra.
A · 0 = 0andA + 1 = 1
ANDing anything with 0 destroys it (the result is 0). ORing anything with 1 forces the result to 1.
A · A = AandA + A = A
A variable ANDed with itself, or ORed with itself, is just itself. (In Boolean algebra, repeating a value does not amplify it.)
A · ¬A = 0andA + ¬A = 1
A variable ANDed with its own negation is always 0 (they can never both be 1 at the same time). A variable ORed with its own negation is always 1 (one of them is always 1).
¬¬A = A
Negating twice returns the original value. After applying De Morgan's Law you often get double NOTs that need cleaning up.
A · B = B · AandA + B = B + A
The order of inputs does not matter. (Just like 2 × 3 = 3 × 2.)
(A · B) · C = A · (B · C)and(A + B) + C = A + (B + C)
For a chain of all-ANDs or all-ORs, the grouping does not matter.
A · (B + C) = A · B + A · CandA + (B · C) = (A + B) · (A + C)
The first is the same as in ordinary algebra (multiplication distributes over addition). The second is unique to Boolean algebra: ordinary + does not distribute over ×, but in Boolean algebra OR distributes over AND.
A + (A · B) = AandA · (A + B) = A
If the same variable already dominates the expression, the other factor is absorbed (it does not change the result). These laws are very common in simplification questions.