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Boolean Algebra

Digital logic works on two values: 1 (true) and 0 (false). Logic gates combine these values; Boolean algebra lets you rewrite and simplify the resulting expressions so circuits use fewer gates.

Showing OCR notationAQA notation: AND = A * B, OR = A + B, NOT = ~A, XOR = A ^ B. Switch board in the header to change it everywhere.

The logic gates

Both boards require AND, OR, NOT, XOR, NAND and NOR.

AND A * B

Output 1 only when both inputs are 1.

ABOut
000
010
100
111

OR A + B

Output 1 when either input is 1.

ABOut
000
011
101
111

NOT ~A

Inverts the single input.

AOut
01
10

XOR A ^ B

Output 1 when inputs differ.

ABOut
000
011
101
110

NAND ~(A * B)

AND followed by NOT (note the bubble).

ABOut
001
011
101
110

NOR ~(A + B)

OR followed by NOT.

ABOut
001
010
100
110

Boolean laws (identities)

Use these to rewrite and simplify expressions. OCR explicitly names De Morgan, distribution, association, commutation and double negation; the rest are the standard identities you are still expected to apply.

Commutation

A * B = B * AA + B = B + A

Association

A * (B * C) = (A * B) * CA + (B + C) = (A + B) + C

Distribution

A * (B + C) = (A * B) + (A * C)A + (B * C) = (A + B) * (A + C)

Identity

A * 1 = AA + 0 = A

Null / annihilation

A * 0 = 0A + 1 = 1

Idempotent

A * A = AA + A = A

Complement (negation)

A * ~A = 0A + ~A = 1

Double negation

~(~A) = A

Absorption

A * (A + B) = AA + (A * B) = A

De Morgan's laws

~(A * B) = ~A + ~B~(A + B) = ~A * ~B

Worked simplification

Simplify ~(~A + B) + A * B:

  1. ~(~A + B) + A * B  — start
  2. (~(~A) * ~B) + A * B  — De Morgan on the bracket
  3. (A * ~B) + A * B  — double negation
  4. A * (~B + B)  — distribution (factor out A)
  5. A * 1  — complement (~B + B = 1)
  6. A  — identity
Exam tip. Always state the law you used at each step — marks are awarded for the named justification, not just the final answer.