Boolean Algebra Axioms Flashcards

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.