Boolean Algebra Theorems Flashcards

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Questions and Answers

What is the simplification rule?

A (A'B) = AB; A+(A'B) = A+B

What does the absorption rule state?

A (A+B) = A; A+(AB) = A

What is the distributive property in Boolean algebra?

A (B+C) = (AB)+(AC); A+(BC)= (A+B)(A+C)

What is DeMorgan's theorem?

<p>A'B' = A' + B'; A'+B' = A'B'</p> Signup and view all the answers

What does the union operation state?

<p>A+1 = 1; A+0 = A</p> Signup and view all the answers

What is the null element?

<p>A*0 = 0; A+1 = 1</p> Signup and view all the answers

What is the complementary rule?

<p>A * A' = 0; A + A' = 1</p> Signup and view all the answers

What is the involution rule?

<p>A'' = A</p> Signup and view all the answers

What defines the idempotent law?

<p>A * A = A; A + A = A</p> Signup and view all the answers

What does the associative law state?

<p>A(BC) = ABC; A+(B+C) = A+B+C</p> Signup and view all the answers

What is the commutative law?

<p>AB = BA; A+B = B+A</p> Signup and view all the answers

What is the identity element for addition and multiplication?

<p>A+0 = A; A*1 = A</p> Signup and view all the answers

What signifies idempotency in Boolean algebra?

<p>If both inputs are identical, the output matches the input.</p> Signup and view all the answers

What does NAND gate represent algebraically?

<p>(A*B)'</p> Signup and view all the answers

What is a maxterm in Boolean algebra?

<p>A Boolean expression resulting in 0 for the output of a single cell.</p> Signup and view all the answers

What is a minterm?

<p>A Boolean expression resulting in 1 for the output of a single cell.</p> Signup and view all the answers

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Study Notes

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.

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