EE 2372 Modern Digital System Design Section 04 Boolean Functions Boolean Functions A function in numerical algebra is an operation that results in an output value for each input value. Boolean Functions In like manner a Boolean function is an operation that results in a logical value (0 or 1) for each set of input logical values (0 or 1). Boolean Functions Consider the function of n variables: Boolean Functions Boolean Functions Boolean Functions These functions may be written out algebraically: Boolean Functions These functions may be written out algebraically: Boolean Functions These functions may be written out algebraically: Boolean Functions These functions may be written out algebraically: Boolean Functions Truth Tables may be used to define switching functions: Boolean Functions Algebraic Forms of Switching Functions "Sum-of-Products" (SOP) Form: (Not really the sum of products, but it resembles a sum of products). Example: Algebraic Forms of Switching Functions "Product-of-Sums" (POS) Form: (Not really the product of sums, but it resembles a product of sums). Example: Algebraic Forms of Switching Functions Canonical Forms are SOP or POS forms with special characteristics. Minterms: Given a function of n variables, if a product term (in the SOP form) contains each variable exactly once (in complemented or uncomplemented form), the product is called a minterm. Algebraic Forms of Switching Functions If a function contains only minterms, it is said to be in Canonical (or Standard) SOP form. A function may be expanded to Canonical SOP form. Algebraic Forms of Switching Functions The switching function may be verified with a truth table. Algebraic Forms of Switching Functions Algebraic Forms of Switching Functions Algebraic Forms of Switching Functions Algebraic Forms of Switching Functions minterms for three variables Algebraic Forms of Switching Functions Maxterms: Given a function of n variables, if a sum term (in the POS form) contains each variable exactly once (in complemented or uncomplemented form) the sum is called a Maxterm. Algebraic Forms of Switching Functions If a function contains only Maxterms, it is said to be in Canonical (or Standard) POS Form. A function may be expanded to Canonical POS Form. Algebraic Forms of Switching Functions A function may expanded to Canonical POS form: Algebraic Forms of Switching Functions A function may expanded to Canonical POS form: The switching function may be verified with a truth table. The switching function may be verified with a truth table. Algebraic Forms of Switching Functions Example Algebraic Forms of Switching Functions Example Algebraic Forms of Switching Functions Maxterms for three variables Algebraic Forms of Switching Functions Algebraic Forms of Switching Functions Algebraic Forms of Switching Functions Incompletely Specified Functions: On occasion, a switching function may include optional minterms or Maxterms. These are "don't-care" conditions. Such conditions may aid in simplifying the functions. "Don't-care" conditions occur when not all combinations of variables are present in a switching function. Incompletely Specified Functions: Example: Incompletely Specified Functions: A switching function that performs the same action (although it is not mathematically equivalent) may be derived by including the first "don't-care" term: