Boolean Algebra Axioms Flashcards

Study Notes

Boolean Algebra Axioms and Properties

Commutative Law: The order of multiplication does not affect the result; expressed as xy = yx.
Associative Law: The way variables are grouped in multiplication does not change the result; represented as x(yz) = (xy)z.
Distributive Law: Multiplication distributes over addition; formulated as x(y + z) = xy + xz.

Fundamental Properties

Absorption Law: Simplifies expressions; stated as x + xy = x, meaning x absorbs the term xy.
DeMorgan's Theorem: Provides a way to simplify negations in products; expressed as !(xy) = !x + !y.

Additional Boolean Expressions

Simplification: The expression x + !xy simplifies to x + y.
Consensus Theorem: Reduces expressions involving products; xy + yz + !xz simplifies to xy + !xz.
Identity with Complement: The product of a variable and its complement always equals zero; verified by x!x = 0.
Identity with One: The sum of any variable and one equals one; shown as x + 1 = 1.
Identity with Zero: The sum of any variable and zero equals the variable itself; shown as x + 0 = x.

Idempotent Properties

Idempotent Law: A variable added to itself remains unchanged; defined as x + x = x.
Complement Law: A variable added to its complement equals one; stated as x + !x = 1.

Combining Variables

Combining Law: A product and its complemented counterpart simplifies to the original variable; noted as xy + x!y = x.

Description

Test your knowledge of the fundamental axioms of Boolean algebra with these flashcards. Each card highlights an essential property, giving you the definitions and examples to master Boolean operations. Perfect for students and anyone looking to reinforce their understanding of abstract algebra concepts.