Boolean algebra revision notes

Purpose of Boolean algebra

Boolean algebra manipulates expressions whose values are either true or false. The main operators are AND, written as · or by placing terms together, OR, written as +, and NOT, written as ¬ or with an apostrophe. Simplification changes the form of an expression without changing its Boolean result.

Useful Boolean identities

Important identities include:

• Identity: A + 0 = A and A · 1 = A
• Domination: A + 1 = 1 and A · 0 = 0
• Idempotent: A + A = A and A · A = A
• Complement: A + ¬A = 1 and A · ¬A = 0
• Double negation: ¬(¬A) = A
• Commutative: A + B = B + A and A · B = B · A
• Associative: (A + B) + C = A + (B + C) and (A · B) · C = A · (B · C)
• Distributive: A · (B + C) = (A · B) + (A · C) and A + (B · C) = (A + B) · (A + C)

The operator symbols must not be treated as ordinary arithmetic. For example, A + A = A in Boolean algebra because OR-ing a value with itself does not alter it.

De Morgan’s laws

De Morgan’s laws move a NOT operation across a group of terms while changing the operator:

• ¬(A · B) = ¬A + ¬B
• ¬(A + B) = ¬A · ¬B

The first law changes NOT-AND into OR of the negated terms. The second changes NOT-OR into AND of the negated terms. Parentheses are important because the NOT applies to the whole grouped expression before the transformation.

Worked simplification

Simplify A + (A · B).

Using the distributive identity in the reverse direction:

A + (A · B) = A · 1 + A · B

Factor out A:

A · 1 + A · B = A · (1 + B)

Using domination, 1 + B = 1, so:

A · (1 + B) = A · 1 = A

Therefore, A + (A · B) = A.

Common errors

Do not remove a NOT without changing the operator when applying De Morgan’s laws. For example, ¬(A + B) is not ¬A + ¬B; it is ¬A · ¬B. Also check the scope of every pair of parentheses, distinguish A + A from A + ¬A, and apply one identity at a time so that each step can be justified.