Boolean Algebra Summary

Discrete Math · Axiom Academy

Let's review how Boolean algebra provides the mathematical foundation for digital circuits, from basic operations to complex sequential systems. AND (·): Output is 1 only when all inputs are 1. Models switches in series. OR (+): Output is 1 when at least one input is 1. Models switches in parallel. NOT ('): Inverts the input. If input is 0, output is 1; if input is 1, output is 0. XOR (⊕): Exclusive OR. Output is 1 when inputs differ. Used in arithmetic circuits and error detection. NAND and NOR: Universal gates that can implement any Boolean function. Identity Laws: A + 0 = A and A · 1 = A Complement Laws: A + A' = 1 and A · A' = 0 Idempotent Laws: A + A = A and A · A = A Absorption Laws: A + AB = A and A(A + B) = A De Morgan's Laws: (A + B)' = A' · B' and (AB)' = A' + B' Duality Principle: Every Boolean identity has a dual obtained by swapping AND ↔ OR and 0 ↔ 1 Gate Symbols: Standard symbols represent Boolean operations in circuit diagrams. Propagation Delay: Time for output to respond to input changes. Critical for timing analysis. Fan-out: Maximum number of gates an output can drive while maintaining correct logic levels. Universal Gates: NAND and NOR can each implement NOT, AND, and OR, making them sufficient for any circuit. Karnaugh Maps: Visual method for finding minimal sum-of-products expressions by grouping adjacent 1s. Prime Implicants: Maximal groups of 1s in K-maps. Essential prime implicants must appear in minimal form.

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