Boolean Algebra Axioms Flashcards

Questions and Answers

What is the definition of the term Commutative in Boolean Algebra?

xy = yx (correct)

x(yz) = (xy)z

x(y + z) = xy + xz

x + xy = x

What is the definition of the term Associative in Boolean Algebra?

x(y + z) = xy + xz

x + xy = x

xy = yx

x(yz) = (xy)z (correct)

What is the definition of the term Distributive in Boolean Algebra?

x + xy = x

x(yz) = (xy)z

xy = yx

x(y + z) = xy + xz (correct)

What does the Absorption law in Boolean Algebra state?

x + xy = x Signup and view all the answers

What is DeMorgan's Theorem?

!(xy) = !x + !y Signup and view all the answers

What is the result of the expression x + !xy?

x + y Signup and view all the answers

What does the Consensus theorem state?

xy + yz + !xz = xy + !xz Signup and view all the answers

What is the value of x!x?

0 Signup and view all the answers

What is the value of x + 1?

1 Signup and view all the answers

What is the value of x + 0?

x Signup and view all the answers

What is the value of x + x?

x Signup and view all the answers

What is the value of x + !x?

1 Signup and view all the answers

What is the result of combining xy + x!y?

x Signup and view all the answers

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.