BOOLEAN ALGEBRA

"A Boolean algebra is a mathematical structure that is similar to a Boolean ring, but that is defined using the meet and join operators instead of the usual addition and multiplication operators."

BASICS

• 1 = TRUE, 0 = FALSE
• Existence of something versus nothing, respectively
• Based on binary system
• Used in digital electronics, set theory, and statistics
• Deals with bits instead of numbers ("truth values")

OPERATIONS

• NOT (if/then), AND (*), OR (+), in that order
• NOT = opposite of given value
• AND = returns 0 unless both values are 1
• OR = returns 1 if at least one value is 1
• ex: Venn diagrams, union, intersection

Untitled Slide

ADDITIVE IDENTITIES

• A + 0 = A
• A + 1 = 1 (1 overrides)
• A + A = A
• A + !A = 1 (complement)

MULTIPLICATIVE IDENTITIES

• 0*A = 0
• 1 * A = A
• A * A = A
• A * !A = 0

BOOLEAN ALGEBRAIC PROPERTIES

• Commutative: A+B = B+A, AB = BA
• Associative: A+(B+C) = (A+B)+C, A(BC)=(AB)C
• Distributive: A(B+C) = AB + AC
• Applies to simplification: less gates = better
• ex: A+AB, A+(!A)B, (A+B)(A+C)

DEMORGAN'S THEOREM

• !(AB) = !(A) + !(B) or (AB)' = A' + B'
• Long bar = grouping symbol
• When bar is broken, operation beneath changes
• Can only break one bar at a time
• ex: !(A+!(BC)), !(AB + CD)

LAST WORDS

• Severely limited in scope
• Laws differ from regular algebra
• Restricted to 1 bit (v. binary)
• Transistors instead of relays now
• Applications to real life