Topic 8 of 8 · Form 5
Boolean algebra and logic circuits
AND, OR, NOT, NAND, NOR and XOR, with truth tables. NAND and NOR are universal gates – any circuit can be built from them alone.
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Logic gates
AND, OR, NOT, NAND, NOR and XOR, with truth tables. NAND and NOR are universal gates – any circuit can be built from them alone.
Boolean algebra laws
- Identity: A + 0 = A; A · 1 = A. Null: A + 1 = 1; A · 0 = 0.
- Idempotent: A + A = A; A · A = A. Complement: A + A′ = 1; A · A′ = 0.
- Absorption: A + A·B = A. Distributive: A·(B + C) = A·B + A·C.
- De Morgan's laws: (A + B)′ = A′·B′ and (A·B)′ = A′ + B′.
Worked example: Simplify A·B + A·B′.
A·(B + B′) = A·1 = A
Karnaugh maps
Group 1s in powers of two (1, 2, 4, 8) to find a minimal sum-of-products expression.
Combinational and sequential circuits
- Half adder: Sum = A XOR B; Carry = A AND B. Full adder combines two half adders with an OR gate.
- Flip-flops (SR, JK, D) store one bit: the basis of registers and memory.
Sum of products
Write an expression from a truth table by ORing the AND-terms for each row with output 1.
Exam tip: Check simplifications by comparing truth tables of the original and simplified expressions.
Check yourself
- Write De Morgan's law for (A·B)′.
- Simplify A + A·B.
- What are the outputs of a half adder?
Show answers
- A′ + B′
- A (absorption law)
- Sum (A XOR B) and Carry (A AND B).
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