Boolean Algebra Theorems Flashcards

Boolean Algebra Theorems and Concepts

• Simplification:

  • A (A'B) simplifies to AB.
  • A + (A'B) simplifies to A + B.

• Absorption:

  • A (A + B) reduces to A.
  • A + (AB) simplifies to A.

• Distributive Law:

  • A (B + C) equals (AB) + (AC).
  • A + (BC) equals (A + B)(A + C).

• DeMorgan's Theorem:

  • The expression A'B' transforms to A' + B' (AND to OR).
  • The expression A' + B' transforms to A'B' (OR to AND).

• Union:

  • A + 1 results in 1.
  • A + 0 equals A.

• Null Law:

  • A * 0 equals 0.
  • A + 1 equals 1.

• Complementary Law:

  • A * A' results in 0.
  • A + A' equals 1.

• Involution Law:

  • A'' simplifies back to A.

• Idempotent Law:

  • A * A equals A.
  • A + A equals A.

• Associative Law:

  • A(BC) simplifies to ABC.
  • A + (B + C) simplifies to A + B + C, demonstrating that grouping does not affect results.

• Commutative Law:

  • AB equals BA.
  • A + B equals B + A, showing order does not matter.

• Identity Law:

  • A + 0 equals A.
  • A * 1 equals A, indicating the identity elements for addition and multiplication.

• Inverse Law:

  • Identifies that the inverse element produces the identity; 0 for OR and 1 for AND.

• Idempotency:

  • Identical inputs in OR or AND gates yield the original value, indicating redundancy.

Logic Gates

• NAND Gate:

  • The operation is represented by (A * B)'.

• OR Gate with Inverted Inputs:

  • Expressed as A' + A'.

• AND Gate with Inverted Inputs:

  • Written as A' * B'.

• NOR Gate:

  • Represented by (A + B)'.

Canonical Forms

• Sum-of-Products (SOP):

  • Consists solely of maxterms, representing a disjunction (OR) of minterms.

• Product-of-Sums (POS):

  • Composed only of minterms, indicating a conjunction (AND) of maxterms.

Minterms and Maxterms

• Minterm:

  • A Boolean expression that results in an output of 1 for a single cell in a Karnaugh map or truth table, outputs 0 elsewhere.

• Maxterm:

  • A Boolean expression that results in an output of 0 for a single cell in a Karnaugh map or truth table, outputs 1 elsewhere.

Test your knowledge of Boolean Algebra with these flashcards highlighting key theorems and simplifications. Each card presents a fundamental concept such as simplification, absorption, and DeMorgan's laws. Perfect for students and professionals looking to strengthen their understanding of digital logic.