1. Min terms and logic expression
A min (product) term is a unique combination of variables
It has a value of 1 for only one input combination
It is 0 for all the other combinations of variables
To write an expression, we need not write the entire truth table
We only need those combinations for which function output is 1
For example, if a function of three variable x, y, and z produces a 1 output for xyz=010, 100, and 111, then it can be written as
f = x’yz’+xy’z’+xyz
We can also treat an order of variables to represent an integer
Thus the function produces a 1 for input values 2, 4, and 7
Some terms may not matter (this is don’t care condition)
Therefore, the function can be written as f(x,y,z)=

2. Max terms and logic expression
A max (sum) term is also a unique combination of variables
However, it is opposite of a min term
It has a value of 0 for only one input combination
It is 1 for all the other combinations of variables
That is why it is called a max (sum) term
Each row in truth table has a max term corresponding to it
Example, a max term (x+y+z) is 0 for combination xyz=000 only
A function can also be written in terms of max terms
The function is product of all max terms for which function is 0
For example, the same function of three variable x, y, and z produces 0 for xyz=000, 001, 011, 101, 110, then
f = (x+y+z).(x+y+z’).(x+y’+z’).(x’+y+z’).(x’+y’+z)
The function can also be written as f(x,y,z)=

3. Don’t care conditions
In some functions certain input combinations can never occur
A common example is representation of a decimal digits
We have 10 digits and thus require 4 bits to encode them
It is called binary-coded decimal
Combinations 0 to 9 are used to code the digits
Combinations never occur
Any output where input is need not be specified
However, this fact can be used to simplify logic function (we will later on how?)
Such don’t care conditions are part of function specification
They can be used for both sum-of-product and product-of-sum forms of functions

4. Adder truth tables and functions
S = A’B’Cin + A’BCin’ + AB’Cin’ + ABCin
Cout = A’BCin + AB’Cin + ABCin’ + ABCin

5. A truth table from a problem statement
A farmer has a dog, a goat, and a cabbage. The goat can eat cabbage and the dog can eat the goat. However, the dog does not like the cabbage. The farmer needs to cross a river and it can only carry two of the three (dog, goat, and cabbage) at a time in the boat. Let variables x, y, and z denote the position of the dog, goat, and cabbage on the south side of the river (0) or on the north side of the river (1), respectively. Let function F be an output of a logic function that will warn the farmer of anything is in danger. Write the truth table for the function.
Let variable w denote the farmer position on the south (0) or the north side (1) of the river. Nothing is in danger in the presence of the farmer. Now modify the truth table for the four variable function.

6. Farmer’s example truth table
With three variables, we do not want X and Y or Y and Z to be equal (both 0 or both 1) at the same time
With four variables, it is not a problem if W is also the same

7. Another function example
A function with two output values, X and Y and four inputs A, B, C, D
X Y together indicate the position of first 1 in listing of inputs
We also count the position from left
What should be the output for ABCD=0000?
Any output is wrong
We need more output combinations
Alternately, we may not care for this input, or have don’t care condition

8. Seven-segment display (multiple outputs)
A function to display digits
4 inputs X, Y, Z, W
Seven outputs
a, b, c, d, e, f, and g
Write functions a, b, c, d, e, f, and g a b c d e f g

9. Seven-segment display (with don’t care)
In seven segment display if we only needs digits 0-9, inputs A-F are don’t care
They can specified accordingly
This fact can be used to simplify function a b c d e f g