2.1 Minterms and matlab calculations  (Page 28/6)

The concepts and procedures in this unit play a significant role in many aspects of the analysis of probability topics and in the use of MATLAB throughout this work.

Minterm vectors and matlab

The systematic formulation in the previous module Minterms shows that each Boolean combination, as a union of minterms, can be designated by a vector of zero-one coefficients. Acoefficient one in the i th position (numbering from zero) indicates the inclusion of minterm M _i in the union. We formulate this pattern carefully below and show how MATLAB logical operations may be utilized in problem setup and solution.

Suppose E is a Boolean combination of $A,B,C$ . Then, by the minterm expansion,

where M _i is the i th minterm and J _E is the set of indices for those M _i included in E . For example, consider

We may designate each set by a pattern of zeros and ones $(e_0,e_1,\cdots ,e_7)$ . The ones indicate which minterms are present in the set. In the pattern for set E , minterm M _i is included in E iff $e_i=1$ . This is, in effect, another arrangement of the minterm map. In this form, it is convenient to view thepattern as a minterm vector , which may be represented by a row matrix or row vector $[e_0{e}_1\cdots e_7]$ . We find it convenient to use the same symbol for the name of the event and for the minterm vector or matrix representing it . Thus, for the examples above,

It should be apparent that this formalization can be extended to sets generated by any finite class.

Minterm vectors for Boolean combinations

If E and F are combinations of n generating sets, then each is represented by a unique minterm vector of length 2 ⁿ . In the treatment in the module Minterms , we determine the minterm vector with the aid of a minterm map. We wish to develop asystematic way to determine these vectors.

As a first step, we suppose we have minterm vectors for E and F and want to obtain the minterm vector of Boolean combinations of these.

1. The minterm expansion for $E\cup F$ has all the minterms in either set. This means the j th element of the vector for $E\cup F$ is the maximum of the j th elements for the two vectors.
2. The minterm expansion for $E\cap F$ has only those minterms in both sets. This means the j th element of the vector for $E\cap F$ is the minimum of the j th elements for the two vectors.
3. The minterm expansion for E ^c has only those minterms not in the expansion for E . This means the vector for E ^c has zeros and ones interchanged. The j th element of E ^c is one iff the corresponding element of E is zero.

We illustrate for the case of the two combinations E and F of three generating sets, considered above