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Boolean simplification preserves a truth function while changing its expression. Name the law at each step and keep logical notation distinct from ordinary arithmetic.
What you'll be able to do
In common Boolean algebra notation, + means OR, multiplication or adjacency means AND, and a bar or NOT means inversion. These are not ordinary arithmetic operations: A+A=A and A×A=A for a Boolean value. Identity laws include A+0=A and A×1=A.
Complement laws give A+NOT A=1 and A×NOT A=0. Double negation returns the original value. Use parentheses where scope could be ambiguous, and state the notation if plain text or a programming language uses different symbols.
De Morgan’s laws exchange AND and OR when negating a group: NOT(A AND B)=(NOT A) OR (NOT B), and NOT(A OR B)=(NOT A) AND (NOT B). Both the operator and every relevant input are inverted.
Distribution allows A(B+C)=AB+AC. Absorption gives A+AB=A: whenever AB is true, A is already true, so the extra term adds no new true rows. These rules justify removing redundant gates while keeping the same function.
Simplification should preserve every input combination, not just the input currently being tested. Produce truth tables for the original and candidate expressions if a transformation is uncertain. A single differing row disproves equivalence.
A shorter expression can reduce hardware resources, but actual circuit design also considers timing, available gate types and shared terms. At this level, first establish logical correctness. Do not cancel Boolean terms using ordinary algebraic division or assume A+1 behaves like numeric addition.
| Law | Example |
|---|---|
| Idempotent | A+A=A |
| Absorption | A+AB=A |
| Complement | A+NOT A=1 |
| De Morgan | NOT(AB)=NOT A + NOT B |
Attempt the task before opening its worked response. Record each change of state, explain the rule used, and check the result against the original requirements. Code examples labelled Python 3 use that language’s conventions; pseudocode states its own assumptions.
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Key takeaways
Test yourself
Answer a few questions, then review anything you missed.
Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.