Min terms and logic expression

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 10-15 never occur • Any output where input is 10-15 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. a f b g e c d 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

9. a f b g e c d 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