$\require{cancel}$
Minterms
 Our ultimate goal is to have sufficient skills to build circuits to accomplish our tasks.
 Frequently we will start with a truth table.
 Two input problems are trivial
 So we will move to three input problems.
 I have a board of three, I want a circuit to detect when a majority vote is achieved.
 So I want the circuit to light up when two or more of the board votes.
 We can assume that they will vote by pressing a button.
 We could mess around and try to build a circuit but let's start with a truth table.
 There are 3 inputs, so $2^3=8$ rows.

a  b  c  Pass 
0  0  0  
0  0  1  
0  1  0  
0  1  1  
1  0  0  
1  0  1  
1  1  0  
1  1  1  
 In chapter 2, Carpinelli discusses Sum of Products (page 213)
 We will discuss thoroughly in Logic and Switching Theory
 We will casually discuss this now in both.
 Given a truth table, we can build an equation by looking at the terms that have a final value of 1.
 these are called the minterms.

a  b  c  Pass 
0  0  0  0 
0  0  1  0 
0  1  0  0 
0  1  1  1 
1  0  0  0 
1  0  1  1 
1  1  0  1 
1  1  1  1 
 Consider a=0, b=1 and c=1.
 f(0,1,1) = 1 if
f(0,1,1) = 0 · 1 · 1
 What is the output of f(a,b,c) = abc?
 We could build a truth table:

a 
b 
c 
a 
ab 
(ab)c) 
0  0  0  1  0  0 
0  0  1  1  0  0 
0  1  0  1  1  0 
0  1  1  1  1  1 
1  0  0  0  0  0 
1  0  1  0  0  0 
1  1  0  0  0  0 
1  1  1  0  0  0 
 Write the similar expressions for the other minterms.
 If we add those together we get
 f(a,b,c) = abc + abc + abc + abc
 From this we can build a circuit
 Build a circuit that detects when any two vote against.
 In Logic and switching we will discuss methods to optimize these circuits.