switching function
a switching function 
is a correspondence that associates an element of the algebra with each of the 
combinations of variables 
. this correspondencee is best specified by means of a truth table. note that each truth table defines only one switching function, although this function may be expressed in a number of ways.
[cite:@kohavi_switching_2010 chapter 3.2 switching functions]
is a correspondence that associates an element of the algebra with each of the combinations of variables . this correspondencee is best specified by means of a truth table. note that each truth table defines only one switching function, although this function may be expressed in a number of ways.[cite:@kohavi_switching_2010 chapter 3.2 switching functions]
the complement 
is a function whose value is 1 whenever the value of is 0, and 0 whenever the value of 
is 1. the sum of two functions and 
is 1 for every combination in which either or 
or both equal 1, while their product is equal to 1 iff both and equal 1. if a function is specified by means of a truth table, its complement is obtained by complementing each entry in the column headed . new functions that are equal to the sum 
and the product 
are obtained by adding or multiplying the corresponding entries in the and columns.
[cite:@kohavi_switching_2010 chapter 3.2 switching functions]
is a function whose value is 1 whenever the value of is 0, and 0 whenever the value of is 1. the sum of two functions and is 1 for every combination in which either or or both equal 1, while their product is equal to 1 iff both and equal 1. if a function is specified by means of a truth table, its complement is obtained by complementing each entry in the column headed . new functions that are equal to the sum and the product are obtained by adding or multiplying the corresponding entries in the and columns.[cite:@kohavi_switching_2010 chapter 3.2 switching functions]
 |  |  | |  | |  |  |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |
and are switching functions with 
variables, we say covers g and write 
if
if and is a multiplication of variables then is called an implicant of and we write 
if and 
then 
and then 
if and only if and 
1st side, assume and , we need to prove
assume that
, then 
is a consequence of that 
, and 
is a consequence of so that , and so
2nd side, assume that
if then and so and so
if then and so and so
and , we need to prove assume that
, then is a consequence of that , and is a consequence of so that , and so 2nd side, assume that
if
then and so and so if
then and so and so a list of switching functions 
of the two variables 
[cite:@kohavi_switching_2010 table 3.6]
of the two variables  |  |  |  | | name of function | symbol |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | inconsistency | |
| 0 | 0 | 0 | 1 |  | nor |  |
| 0 | 0 | 1 | 0 |  | ||
| 0 | 0 | 1 | 1 |  | not | |
| 0 | 1 | 0 | 0 |  | ||
| 0 | 1 | 0 | 1 |  | ||
| 0 | 1 | 1 | 0 |  | xor |  |
| 0 | 1 | 1 | 1 |  | nand |  |
| 1 | 0 | 0 | 0 |  | and |  |
| 1 | 0 | 0 | 1 |  | equivalence |  |
| 1 | 0 | 1 | 0 |  | ||
| 1 | 0 | 1 | 1 |  | material implication |  |
| 1 | 1 | 0 | 0 |  | ||
| 1 | 1 | 0 | 1 |  | implication |  |
| 1 | 1 | 1 | 0 |  | or | |
| 1 | 1 | 1 | 1 | 1 | tautology |
finding the switching expressions of a switching function
we write a truth table for the function and for each row we write a minterm or a maxterm depending on whether the function outputs 1 or 0, respectively
| x | y | z | f | minterms | maxterms | |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 |  | |
| 1 | 0 | 0 | 1 | 0 |  | |
| 2 | 0 | 1 | 0 | 1 |  | |
| 3 | 0 | 1 | 1 | 1 |  | |
| 4 | 1 | 0 | 0 | 0 |  | |
| 5 | 1 | 0 | 1 | 0 |  | |
| 6 | 1 | 1 | 0 | 1 |  | |
| 7 | 1 | 1 | 1 | 1 |  |
is a sum of products, this is called the canonical sum of products or disjunctive normal form
is a product of sums, called the canonical product of sums or conjunctive normal formthe process of finding a minimal switching expression of a switching function
given the switching function
- find all the broken link: blk:1707154256s of
- from the list of prime implicants, find the essential prime implicants and put them in an expression
- remove from the list of PI all the EPI's and all the PI's that are covered by EPI's
- if the set of EPI's covers , then we're done, otherwise, choose more PI's so that is entirely covered by the resulting set of PI's but make sure to pick a minimal number of PI's to do the job, and make sure to pick the PI's with the least number of literals
this isnt a definite solution for the problem but its enough for functions with a small number of literals
using karnaugh maps we can apply this processeach cell maps to a combination of variables, we place each minterm in its corresponding cell, i.e. with the cell that provides the correct corresponding values for the variables
by finding rectangles whose area is a power of 2 in these tables we would be finding minterms and the least amount of rectangles we can find would give us the minimal form we're looking for
the variables that cells in a rectangle may share are the ones to be taken for the switching expression