Boolean algebra is a system of mathematics built around values that are either 0 or 1, often interpreted as false or true. It matters because digital computers, calculators, phones, and control systems all use circuits that follow Boolean rules. Instead of measuring continuously changing quantities, Boolean algebra describes decisions, switches, and logical conditions.
This makes it the mathematical language of digital logic.
Key Facts
- Boolean variables usually take only two values: 0 and 1.
- AND operation: A · B = 1 only when A = 1 and B = 1.
- OR operation: A + B = 1 when A = 1, B = 1, or both are 1.
- NOT operation: A' is the complement of A, so if A = 1 then A' = 0.
- De Morgan's laws: (A · B)' = A' + B' and (A + B)' = A' · B'.
- Distributive laws: A · (B + C) = A · B + A · C and A + B · C = (A + B) · (A + C).
Vocabulary
- Boolean variable
- A symbol such as A or B that can have only the value 0 or 1.
- AND
- A Boolean operation that outputs 1 only when all of its inputs are 1.
- OR
- A Boolean operation that outputs 1 when at least one input is 1.
- NOT
- A Boolean operation that reverses a value, changing 1 to 0 and 0 to 1.
- Logic gate
- An electronic circuit element that performs a Boolean operation on input signals.
Common Mistakes to Avoid
- Treating + as ordinary addition is wrong because in Boolean algebra 1 + 1 = 1 for OR, not 2.
- Forgetting operation order is wrong because NOT is applied before AND, and AND is usually applied before OR unless parentheses say otherwise.
- Using De Morgan's laws without changing the operation is wrong because complementing A · B gives A' + B', not A' · B'.
- Assuming A + A' = 0 is wrong because a variable OR its complement is always 1.
Practice Questions
- 1 Evaluate F = A · B + C for A = 1, B = 0, and C = 1.
- 2 Complete the truth table for F = (A + B)' for the four input pairs A B = 00, 01, 10, 11.
- 3 Explain why the expression (A · B)' can be built using an AND gate followed by a NOT gate, or equivalently by using NOT gates on A and B followed by an OR gate.