Boolean simplification exercises with answers: basic
Ten one-step simplifications, one law each: identity, domination, idempotence, complement, absorption and double negation. Name the law, write the result, and compare it with the step list under the problem, where the engine names the same law and shows the rewrite.
These are the laws the simplifier applies on its own, each as a local rewrite that makes the expression smaller: a constant disappears (identity), a constant takes over (domination), a repeated operand collapses (idempotence), a value meets its own negation (complement), a term that contains the other operand is absorbed, and two negations cancel. In every problem the result is verified equivalent on every row of the truth table shown above the steps.
Read the classification line as well. A OR TRUE and A OR NOT A simplify to the constant TRUE and are tautologies; A AND FALSE and A AND NOT A simplify to FALSE and are contradictions; the rest keep a variable and are contingent. Noticing that a rule has collapsed to a constant is usually the most useful thing simplification tells you about it.
1. Simplify A AND TRUE and name the law. A AND TRUE
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | TRUE |
A AND TRUE- Identity ·
A AND TRUE = AA AND TRUE→AResult:A
Classification: contingent
Identity: TRUE is the neutral element of AND, so the conjunction keeps the value of A. The result is A.
2. Simplify A OR FALSE and name the law. A OR FALSE
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | TRUE |
A OR FALSE- Identity ·
A OR FALSE = AA OR FALSE→AResult:A
Classification: contingent
Identity again: FALSE is the neutral element of OR, and the result is A.
3. Simplify A OR TRUE and name the law. A OR TRUE
| # | A | f |
|---|---|---|
| 0 | FALSE | TRUE |
| 1 | TRUE | TRUE |
A OR TRUE- Domination ·
A OR TRUE = TRUEA OR TRUE→TRUEResult:TRUE
Classification: tautology
Domination: TRUE takes over any OR, so the result is the constant TRUE. The table is TRUE on both rows and the classification says tautology.
4. Simplify A AND FALSE and name the law. A AND FALSE
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | FALSE |
A AND FALSE- Domination ·
A AND FALSE = FALSEA AND FALSE→FALSEResult:FALSE
Classification: contradiction
Domination: FALSE fixes any AND, so the result is FALSE, a contradiction.
5. Simplify A OR A OR A and name the law. A OR A OR A
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | TRUE |
A OR A OR A- Idempotence ·
A OR A = AA OR A OR A→AResult:A
Classification: contingent
Idempotence: a value joined with itself by OR is the value, and the simplifier merges the three operands in one step. The result is A.
6. Simplify A OR NOT A and name the law. A OR NOT A
| # | A | f |
|---|---|---|
| 0 | FALSE | TRUE |
| 1 | TRUE | TRUE |
A OR NOT A- Complement ·
A OR NOT A = TRUEA OR NOT A→TRUEResult:TRUE
Classification: tautology
Complement: a value OR its negation is TRUE on every row, so the result is TRUE and the formula is a tautology.
7. Simplify A AND NOT A and name the law. A AND NOT A
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | FALSE |
A AND NOT A- Complement ·
A AND NOT A = FALSEA AND NOT A→FALSEResult:FALSE
Classification: contradiction
Complement: a value AND its negation is FALSE on every row; the result is FALSE and the classification says contradiction.
8. Simplify A OR (A AND B) and name the law. A OR (A AND B)
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | FALSE |
| 1 | FALSE | TRUE | FALSE |
| 2 | TRUE | FALSE | TRUE |
| 3 | TRUE | TRUE | TRUE |
A OR (A AND B)- Absorption ·
A OR (A AND B) = AA OR (A AND B)→AResult:A
Classification: contingent
Absorption: A AND B can only be TRUE when A already is, so the term adds no row and the result is A. The table is TRUE exactly on A = TRUE, B = FALSE and A = TRUE, B = TRUE.
9. Simplify A AND (A OR B) and name the law. A AND (A OR B)
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | FALSE |
| 1 | FALSE | TRUE | FALSE |
| 2 | TRUE | FALSE | TRUE |
| 3 | TRUE | TRUE | TRUE |
A AND (A OR B)- Absorption ·
A AND (A OR B) = AA AND (A OR B)→AResult:A
Classification: contingent
Absorption in its dual form: when A is TRUE, A OR B is TRUE and the AND gives A; when A is FALSE the AND is FALSE. The result is A.
10. Simplify NOT NOT A and name the law. NOT NOT A
| # | A | f |
|---|---|---|
| 0 | FALSE | FALSE |
| 1 | TRUE | TRUE |
NOT NOT A- Double negation ·
NOT NOT A = ANOT NOT A→AResult:A
Classification: contingent
Double negation: the two negations cancel and the result is A.
Questions
Why is the result A and not something shorter?
A single variable is as short as a formula gets. The simplifier reports its output as simplified rather than as the shortest form, because in general its laws do not search for one; on this page every result is a literal or a constant, so there is nothing left to shorten.
Related
Laws
- Idempotent law: A AND A = A, A OR A = A
- Double negation (involution): NOT NOT A = A
- Consensus theorem: AB + A'C + BC = AB + A'C, with proof
Pitfalls
Tools
See also
- Lesson 7: Simplification laws (intermediate, 12 min)
- Identity law of Boolean algebra: A AND TRUE = A, A OR FALSE = A
- Null law (domination): A AND FALSE = FALSE, A OR TRUE = TRUE
- Absorption law: A OR (A AND B) = A, with proof
- Complement law: A AND NOT A = FALSE, A OR NOT A = TRUE
Practice
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