Boolean Algebra · step 5 of 13

Boolean algebra, logic & set algebra

Boolean algebra, propositional logic and set algebra are three notations for one structure. The dictionary is exact: + ↔ ∨ ↔ ∪, · ↔ ∧ ↔ ∩, ′ ↔ ¬ ↔ complement, 1 ↔ ⊤ ↔ U, 0 ↔ ⊥ ↔ ∅. Every theorem proved in one notation is instantly a theorem in the others, which is why De Morgan appears three times in a discrete maths course.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Translate symbol by symbolUse the dictionary above mechanically. The structure of the expression never changes.
Recognise the shared theoremsDe Morgan, distributivity, absorption and idempotence are the same statement in each notation.
Pick the notation that makes the problem easyCounting arguments suit sets; circuit questions suit Boolean; proof questions often suit logic.
Watch for notation-specific extrasLogic has → and ↔ which have no standard Boolean symbol — rewrite them as x′ + y and (x′+y)(y′+x) first.

Worked example

Translate (A ∪ B)ᶜ ∩ C into Boolean algebra and into propositional logic.

∪ becomes +, ᶜ becomes ′, ∩ becomes ·.
Boolean: (a + b)′·c.
For logic, ∪ becomes ∨, complement becomes ¬, ∩ becomes ∧.
Logic: ¬(p ∨ q) ∧ r.

Answer. (a + b)′c in Boolean algebra and ¬(p ∨ q) ∧ r in logic — and De Morgan simplifies all three to a′b′c, ¬p ∧ ¬q ∧ r, Aᶜ ∩ Bᶜ ∩ C.

Where marks get dropped

These are the specific errors that cost credit on boolean algebra, logic & set algebra questions — QED's rubric penalises each of them separately.

Translating → directly. There is no primitive Boolean implication; rewrite p → q as p′ + q first.
Mixing notations within one expression, such as writing A + ¬B. Choose one and stay in it.
Assuming set operations behave differently. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) is the same law as x + yz = (x+y)(x+z).

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Answer in real notation

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Boolean algebra, logic & set algebra — frequently asked questions

Are these really the same structure?

Yes. Each is a Boolean algebra in the axiomatic sense, and Stone’s representation theorem says every Boolean algebra is isomorphic to an algebra of sets.

Which notation should I use in an exam?

The one the question uses. Translating to your favourite is fine for working, but the answer should return to the notation asked for.

Why does the power set appear so often?

𝒫(S) with ∪, ∩ and complement is the model example of a Boolean algebra, and every finite Boolean algebra is isomorphic to one — hence sizes are always powers of 2.

The rest of Boolean Algebra

Axioms, laws, simplification and Boolean functions. Each subtopic below has its own method, worked example and mark-losing traps.

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