ITEC 352 Lecture 4 Boolean logic / Karnaugh Maps

K-Maps / Boolean logic Review Truth tables Conversion between Multiplexers

K-Maps / Boolean logic Outline Multiplexers Math –Karnaugh Maps –Minterm / Maxterm –Relationship to gates

K-Maps / Boolean logic Multiplexers Purpose? Construction using gates

K-Maps / Boolean logic Demultiplexer Only passes a 1 through if the input is 1

K-Maps / Boolean logic Combinatio n Why would you combine a multiplexer and a de-multiplexer?

K-Maps / Boolean logic Canonical forms A Boolean expression can be expressed in a number of ways. –E.g., S = X’+Y*Z can also be represented as S = X’ + X’ + Y*Z Hence,we use canonical forms (or normalized forms) that are unique in representing in expressions. –Note, they need not be minimal – they are just unique. Two popular canonical forms: –Sum of products and Product of sums.

K-Maps / Boolean logic Minterms Minterm: Boolean function that is 1 in only one row of the truth table. –Example: what are the minterms of function s. –S=X’ +Y*Z –Represent the sum (s) function using minterms.

K-Maps / Boolean logic Maxterms Maxterm: Boolean function that is 0 in only one row of the truth table. For example, the function: –Example maxterms for the sum function. –S=X’ +Y*Z Exercise: express F = X + Y' * Z as product of sums (in terms of maxterms)

K-Maps / Boolean logic Class exercise Implement a majority function: –Given three inputs, the functions equals 1, if the majority of inputs are 1’s else it is a 0. Steps: –(a) Draw the truth table. –(b) Write down the function in a canonicalized way. –(c) Draw the circuit.

K-Maps / Boolean logic K-Maps 2 Level Maximum

K-Maps / Boolean logic Constructio n Variant of a truth table –Grid format Allows you to see patterns quickly

K-Maps / Boolean logic Terms Implicant: a single minterm or group of minterms that can be combined together on the K-map Prime-implicants: Implicant that can not be combined with another one to remove a literal Essential prime implicants: prime implicant that includes a minterm not covered by any other prime implicant

K-Maps / Boolean logic Exercise Reduce the following Boolean expression using a Karnaugh Map: F = A’BC + AB’C + ABC + ABC’

K-Maps / Boolean logic Review A few mathematical terms –Minterm –Maxterm Visualization technique for Boolean expressions