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Boolean Algebra Review and Practice

Tutorial Index

Boolean algebra review

The central ideas of this section can be kept surprisingly compact.

Symbols: ~ = NOT    • = AND    + = OR    ⊕ = XOR

Core habits: simplify easy pieces first; use complements; factor when useful; recognize absorption; apply De Morgan carefully.

A small reference sheet

A + 0 = A     A•1 = A
A + 1 = 1     A•0 = 0
A + A = A     A•A = A
A + ~A = 1     A•~A = 0
A + A•B = A
A•(A+B) = A
~(A•B) = ~A + ~B
~(A+B) = ~A•~B

One final bridge to the real circuit

A smaller expression often means a smaller circuit, but smaller is not the only goal. A practical designer also asks:

Can each output drive its loads? Is the noise margin adequate? Will the total propagation delay meet the timing requirement? Are there glitches during transitions? Are the voltage, current, temperature, and power ratings satisfied?

Review problems

Simplify A + A•B.
Absorption gives A + A•B = A.
Simplify A•B + A•~B.
Factor A: A(B+~B)=A•1=A.
Which is De Morgan's equivalent of ~(A•B)?
NOT(A AND B) equals NOT A OR NOT B.
A correct Boolean expression is implemented with too many loads on one output. What kind of problem is this?
The logic can be correct while the electrical loading exceeds the output specification.
A signal must pass through six gates in series. Which real-world quantity may accumulate?
Each stage takes finite time to respond, so propagation delays along a path can accumulate.
Which statement best summarizes Boolean algebra in digital design?
Boolean algebra gives us a mathematical language for two-state logic; real implementation still requires engineering judgment.

Where this leads next

Boolean algebra is now ready to be applied to something very physical: relay contacts and control circuits. Relays make AND, OR, NOT, interlocks, seal-in circuits, and timing relationships visible in a way that is especially useful before moving deeper into digital ICs.

< Expressions and Gate Circuits   |   State and Memory >