ITEC 352

Lecture 4Boolean 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

Combination

• 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

Construction

• 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