Boolean Algebra Rules – Quick Recall Guide with Worked Examples
Introductory Note:
This guide is designed as a quick-recall and summary reference for the fundamental rules of Boolean
algebra. Each rule is presented in a clear, numbered format with a one-sentence explanation for instant
understanding, followed by worked examples that show how the rule is applied in simplifying
expressions. The goal is to make review sessions faster, exam preparation more efficient, and practical
application in logic design or digital circuits easier.
Basic Boolean Operations Refresher
Before applying the rules, it’s important to recall the three core operations used throughout Boolean
algebra:
1. Logical Addition (+) – OR
o Meaning: The result is 1 (true) if at least one input is 1.
o Truth Table:
2. Logical Multiplication (·) – AND
o Meaning: The result is 1 (true) only if both inputs are 1.
o Truth Table:
3. Complement (x̅ ) – NOT
o Meaning: Reverses the value of a variable — if it’s 1, it becomes 0; if it’s 0, it becomes
1.
o Examples:
, Boolean Algebra Rules:
Rule #1: 0 + x = x, adding 0 to any variable with OR doesn’t change its value because 0 represents false
and contributes nothing to the logical outcome.
fast recall, I can help you batch the rest of the core rules—want to move on to Rule No. 2?
Rule #2: 1 + x = 1, any value with 1 always gives 1 because 1 represents true, and once true is present in
an OR operation, the whole expression is true regardless of the other variable.
Rule #3: x + x = x, a variable with itself doesn’t change its value because if x is true, the result is already
true, and if x is false, repeating it doesn’t make it true.
Rule #4: x + x̅ = 1, a variable OR’ed with its complement (x̅, meaning NOT x) is always 1 because one of
them must be true, so the whole expression is guaranteed to be true.
Rule #5: 0 · x = 0, any value with 0 always gives 0 because 0 represents false, and false in an AND
operation overrides everything, making the whole expression false.
Introductory Note:
This guide is designed as a quick-recall and summary reference for the fundamental rules of Boolean
algebra. Each rule is presented in a clear, numbered format with a one-sentence explanation for instant
understanding, followed by worked examples that show how the rule is applied in simplifying
expressions. The goal is to make review sessions faster, exam preparation more efficient, and practical
application in logic design or digital circuits easier.
Basic Boolean Operations Refresher
Before applying the rules, it’s important to recall the three core operations used throughout Boolean
algebra:
1. Logical Addition (+) – OR
o Meaning: The result is 1 (true) if at least one input is 1.
o Truth Table:
2. Logical Multiplication (·) – AND
o Meaning: The result is 1 (true) only if both inputs are 1.
o Truth Table:
3. Complement (x̅ ) – NOT
o Meaning: Reverses the value of a variable — if it’s 1, it becomes 0; if it’s 0, it becomes
1.
o Examples:
, Boolean Algebra Rules:
Rule #1: 0 + x = x, adding 0 to any variable with OR doesn’t change its value because 0 represents false
and contributes nothing to the logical outcome.
fast recall, I can help you batch the rest of the core rules—want to move on to Rule No. 2?
Rule #2: 1 + x = 1, any value with 1 always gives 1 because 1 represents true, and once true is present in
an OR operation, the whole expression is true regardless of the other variable.
Rule #3: x + x = x, a variable with itself doesn’t change its value because if x is true, the result is already
true, and if x is false, repeating it doesn’t make it true.
Rule #4: x + x̅ = 1, a variable OR’ed with its complement (x̅, meaning NOT x) is always 1 because one of
them must be true, so the whole expression is guaranteed to be true.
Rule #5: 0 · x = 0, any value with 0 always gives 0 because 0 represents false, and false in an AND
operation overrides everything, making the whole expression false.
