Boolean Algebra and Logical Functions

Logical Functions

• The NOT operator inverts the input value; true becomes false and vice versa.
• XOR (exclusive OR) is true when inputs differ, represented as x ⊕ y.
• XOR truth table:
  • 0 ⊕ 0 = 0
  • 0 ⊕ 1 = 1
  • 1 ⊕ 0 = 1
  • 1 ⊕ 1 = 0
• XNOR (exclusive NOR) is true when inputs are the same and defined as the negation of XOR, using the operator x ⊙ y.

Fundamental Identities in Boolean Algebra

• Commutative Law: x+y = y+x; x⋅y = y⋅x (order does not matter).
• Associative Law: (x + y) + z = x + (y + z); x⋅(y⋅z) = (x⋅y)⋅z (grouping does not change value).
• Idempotent Law: x+x = x; x⋅x = x (combining same terms yields the same term).
• Annihilator Law: x+1 = 1; x⋅0 = 0 (caps or zeros dominate).
• Identity Law: x+0 = x; x⋅1 = x (zero or one does not change the term).
• Complement Law: x+x' = 1; x⋅x' = 0 (a term paired with its complement yields extremes).
• Absorptive Law: x + xy = x + y (reducing complexity by absorbing terms).
• Distributive Law: x⋅(y + z) = xy + xz; (x + y)⋅(p + q) = xp + xq + yp + yq (distributing factors).
• DeMorgan's Law: Negation of an AND becomes OR of negations, and vice versa: x+y' = x⋅y; x⋅y' = x+y.

Simplification Example

• Simplifying A(A + B) + BA:
  • A(A + B) + BA transforms through Boolean laws to result in A.

Problem-Solving Approaches

• Finding ordered pairs (A, B) that satisfy expressions can involve examining the truth table or applying algebraic simplification methods.### Online Resources for Boolean Algebra

• Ryan's Tutorials offers an extensive online tutorial for Boolean Algebra, accessible at Ryan's Tutorials.

• Multiple Boolean calculators are available online; one notable option provides a Truth Tables calculator, located at Truth Tables Calculator.

• A calculator that simplifies Boolean expressions and includes a NOT operator represented by "!" can be found at Boolean Algebra Calculator.

Video Resources

• ACSL has curated a selection of YouTube videos where students and advisors work through previous problems related to Boolean Algebra.
• Videos can be directly accessed by clicking on their titles within the YouTube platform; users should consider larger views for clarity.
• Some video content may be accompanied by ads, with ACSL not profiting from these advertisements.

Specific Video Content by Mr. Minich

• Worksheet 1 Video: Mr. Minich, an ACSL advisor, explains solutions to five problems from recent years relevant to the ACSL Boolean Algebra category. Watch it at Worksheet 1 Video.
• Worksheet 2 Video: Another video featuring Mr. Minich that provides additional problem-solving techniques for the ACSL Boolean Algebra contest. Accessible at Worksheet 2 Video.