Boolean Algebra Basics

Practice vocabulary flashcards covering the fundamental laws, rules, logic operations, and DeMorgan's theorems in Boolean algebra.

Boolean Algebra

A branch of algebra in which the values of the variables are either true or false, denoted as $1$ and $0$ in digital logic systems.

Boolean Addition

The logic operation equivalent to the OR operation (symbol: $+$), producing a sum term of $1$ if one or more literals are $1$, and $0$ only if every literal is $0$.

Boolean Multiplication

The logic operation equivalent to the AND operation (symbol: $\cdot$ or no symbol), producing a product term that is $1$ only if all literals are $1$.

Commutative Law of Addition

A Boolean algebra law stating that the order of the addition operands does not affect the result: $A + B = B + A$.

Commutative Law of Multiplication

A Boolean algebra law stating that the order of the multiplication operands does not affect the result: $A \cdot B = B \cdot A$.

Associative Law of Addition

A Boolean algebra law stating that grouping does not affect the result of an OR operation: $(A + B) + C = A + (B + C)$.

Associative Law of Multiplication

A Boolean algebra law stating that grouping does not affect the result of an AND operation: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$.

Distributive Law

A Boolean algebra law stating that Boolean multiplication distributes over addition: $A \cdot (B + C) = A \cdot B + A \cdot C$.

Extended Distributive Rule

The rule stating that summing a variable with the product of two other variables yields a product-of-sums (POS): $A + (B \cdot C) = (A + B) \cdot (A + C)$.

Identity Rule

A Boolean rule stating that adding $0$ leaves the variable unchanged ($A + 0 = A$), and multiplying by $1$ leaves the variable unchanged ($A \cdot 1 = A$).

Null Rule

A Boolean rule stating that adding $1$ always results in $1$ ($A + 1 = 1$), and multiplying by $0$ always results in $0$ ($A \cdot 0 = 0$).

Idempotent Rule

A Boolean rule stating that repeating the same variable does not change the result: $A + A = A$ and $A \cdot A = A$.

Involution Rule

A Boolean rule stating that double negation returns the original value of the variable: $(A')' = A$ or $\overline{\overline{A}} = A$.

Complement Rule

A Boolean rule stating that a variable ORed with its complement equals $1$ ($A + A' = 1$), and a variable ANDed with its complement equals $0$ ($A \cdot A' = 0$).

Absorption Rule

A Boolean rule where the original variable absorbs a product or sum term involving itself: $A + (A \cdot B) = A$ and $A \cdot (A + B) = A$.

DeMorgan’s First Theorem (Law of Intersection)

A theorem stating that the complement of a product is equal to the sum of the complements: $(A \cdot B)' = A' + B'$.

DeMorgan’s Second Theorem (Law of Union)

A theorem stating that the complement of a sum is equal to the product of the complements: $(A + B)' = A' \cdot B'$.

Negative-OR Gate

The DeMorgan equivalent of a NAND gate, consisting of an OR gate with inverters (bubbles) on its inputs, implementing the function $X = A' + B'$.

Negative-AND Gate

The DeMorgan equivalent of a NOR gate, consisting of an AND gate with inverters (bubbles) on its inputs, implementing the function $X = A'B'$.

Duality (DeMorgan's Rule)

The process of interchanging operators and constants in a Boolean function by replacing $0$ with $1$, $1$ with $0$, the AND operator with the OR operator, and the OR operator with the AND operator.

Redundancy Rule

The Boolean simplification identity stating that $A + A'B = A + B$ (or $A + \overline{A}B = A + B$).

Exclusive-OR (XOR)

A logic operation denoted by $\oplus$, defined by the Boolean expressions $X \oplus Y = X'Y + XY' = (X + Y) \cdot (X' + Y')$.

Exclusive-NOR (XNOR)

A logic operation denoted by $\odot$, defined by the Boolean expressions $X \odot Y = X'Y' + XY = (X + Y) \cdot (X \cdot Y)'$.