Boolean algebra practice problems
Boolean algebra sticks once you have simplified a few dozen expressions yourself and seen each answer checked. Below are three kinds of problem that never run out: simplify a sum of minterms to its minimal form, pick the expression that is equivalent to a given one, and name the law behind an identity. Every answer is checked against the full truth table, and a simplified answer opens as a live gate circuit.
The laws of Boolean algebra
| Law | OR form | AND form |
|---|---|---|
| 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 |
| Involution | (A')' = A | |
| Commutative | A + B = B + A | A B = B A |
| Associative | (A + B) + C = A + (B + C) | (A B) C = A (B C) |
| Distributive | A + B C = (A + B)(A + C) | A (B + C) = A B + A C |
| Absorption | A + A B = A | A (A + B) = A |
| De Morgan | (A + B)' = A' B' | (A B)' = A' + B' |
| Consensus | A B + A' C + B C = A B + A' C | (A + B)(A' + C)(B + C) = (A + B)(A' + C) |
Each law has two forms that are duals: swap + with AND and 0 with 1, and one becomes the other. Here a space or nothing between letters means AND, + means OR and ' means NOT. Read the full explanations in the Boolean algebra lesson and see De Morgan's laws as gates.
A worked example
Simplify F = A'BC + AB'C + ABC. The last two terms differ only in B, so
AB'C + ABC = AC(B' + B) = AC. The first and last differ only in A, so
A'BC + ABC = BC. A term may be used twice (A + A = A), so both merges are allowed and
F = AC + BC, four literals instead of nine. Check it on a row: A = 0, B = 1, C = 1 gives
1 in both forms. When the groups are hard to see, draw a Karnaugh map,
or let the Boolean algebra calculator show the minimal form.
Typing your answer
Write NOT as a prime after the letter or a bracket (A', (A + B)'), AND by putting letters side by side (AB' or A B'), and OR as +. The words AND, OR, NOT and XOR work too. An answer counts as minimal when it is correct and uses no more literals (letters) than a minimal sum of products.
Frequently asked
How do you practice Boolean algebra?
Simplify many small expressions and check each answer against its truth table. This page generates problems without end, grades a typed answer for both correctness and minimality, and shows a minimal answer when you are stuck.
How do you simplify a Boolean expression?
Look for pairs of terms that differ in exactly one variable and merge them, since AB + AB' = A(B + B') = A. Repeat until nothing merges, then drop any term the others already cover. A Karnaugh map does the same grouping visually.
What are the laws of Boolean algebra?
Identity, null, idempotent, complement, involution, commutative, associative, distributive, absorption, De Morgan's laws and consensus. Each has an OR form and an AND form that are duals of each other.
How do I know my simplified answer is right?
Two expressions are equal exactly when their truth tables match on every row. This page checks all eight rows over A, B and C, and then compares the number of literals with a minimal answer.
Is there more than one minimal answer?
Sometimes. A function can have several minimal sums of products with the same number of literals. Any of them is marked correct here; the page shows one.
Practice the gates themselves with the logic gates practice quiz, turn a truth table into an expression with the truth table to expression converter, or draw any expression as gates with the logic circuit generator. In the free course you build those gates from transistors, up to an 8-bit CPU. For number practice, try the binary game.
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