Minterms & maxterms

applications of boolean algebra

1. Application of boolean algebra (Minterms & Maxterms expansions) Presented By : Nousheen Arshad, Tayyaba Shahzad , Hammad Khan & Hassan Ayaz Kiani

2. : Minterms : A Minterm (product) is a combination of variables : It has a value of 1 for only one input combination. It is 0 for all the other combinations of variables. A Minterm of n variables is the product of n literals in which each variable appears once in either a complemented or uncomplemented form. For example: If a function of 3 variables x,y and z produces as 1 output for xyz=010, 100, 111, then it can be written as F= x'yz'+ xy'z'+ xyz.

3. : Minterms Expansion : If a minterm has a single 1 and the remaining cells as 0's, it would appear to cover a minimum area of 1's. It is standard Sum-of-Products (SOP). F= A'BC+AB'C'+AB'C+ABC has four minterms. This function will be written as F= Σ m(3,4,5,7).

4. : Example :

5. : Maxterms : A Max (sum) term is also a unique combination of variables : However, It is opposite of minterms. 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 Max (sum) terms. For Example: The same function of 3 variables x,y and z produces as 0 output for xyz=000, 001, 011, 101, 110, then F= (x+y+z).(x+y+z').(x+y'+z').(x'+y+z').(x'+y'+z)

6. : MAxterms Expansion : It is a standard Product of Sum (POS). We can also treat an order of variables to represent an integer . The function can also be written as f(x,y,z) = π M(0,1,3,5,6).

7. : Example :

8. : Conversion of English sentences to boolean equations : For simple problems, go directly from a word description of the desired circuit behavior to an boolean expression. For Example: The alarm will ring (Z) iff The alarm is turned on (A), and The door is not closed (B'), or Or, It is after 6 p.m (C), and The window is not closed (D'). Z = AB'+CD'.

9. : Combinational logic design using a truth table : Example no 01: The canonical minterm (SOP) form F= A'B'+A'B. The canonical maxterm (POS) form F= (A'+B).(A'+B').

10. : Example no 02 : Maxterm Expansion: F(A,B,C) = (A+B+C).(A+B+C').(A+B'+C) = M0 + M1+ M2. = π M(0,1,2). Minterm Expansion: F(A,B,C) = A'BC+AB'C'+AB'C+ABC'+ABC. = m3+m4+m5+m6+m7. = Σ m(3,4,5,6,7).

11. : Design of binary adders : Design simple binary adders that adds two-bit binary numbers (N1, N2) and produces 3-bit sum (N3).

12. : Binary Adder : 1 bit binary adder... Inputs: A,B, Carry-In. Output: Sum, Carry-Out. Sum = A'B'Cin + A'BCin' + AB'Cin' + ABCin. Cout = A'BCin + AB'Cin + ABCin' + ABCin. Both sum and cout can be minimized.