ME290M, Spring 1999

ME290M
Expert Systems in Mechanical Engineering
Spring 1999, T-Th 12:30-2:00 pm
1165 Etcheverry Hall, Course Control No. 56369 http://best.me.berkeley.edu/~aagogino/me290m/s99

Multivariate Logic

In multivariate logic, the atomic expressions no longer evaluate to either "true" or "false". In fact, they may evaluate to many events each having a measure of "trueness" associated with it. We will use the concept of sample space to introduce the concept of multiple but well defined events in multivariate logic. A sample space is a generalized collection of points representing elementary events. These elementary events may be conveniently combined into collections or sets of events.

1. Venn Diagrams

Venn diagrams are graphical representations of sample spaces, elementary events, and event sets. Each event is assigned an area in the sample space, corresponding to its proportion of elementary events as shown in Figure 1. In this diagram the event sets are A, B, and C. Because the area of an event in a Venn diagram corresponds to the number of elementary events it contains, Figure 1 implies that event B is a larger set than event A, which is in turn a larger set than event C. We also observe that events A and B share some elementary events in common as do events A and C.

Figure 1: Venn Diagram of the Sample Space "S"

The set of all elementary events is called the universe and is designated as " I". A set which contains no elementary events is called the null set and is designated as "{}".

1.1 Complementation

The complement of an event A is defined as the set of all events in the sample space that does not include any of the elementary events in A. The complement of A is shown in Figure 2 and is designated as A' or an overscored A.

Figure 2: Complementation

1.2 Intersection or Product of Events

Two events, A and B, may intersect and the area of intersection is designated as AB or A´B. The intersection or product of A and B is the collection of all elementary events contained in both A and B as shown in the left half of Figure 3 below.

1.3 Union or Sum of Events

The union of two events A and B is the collection of all elementary events contained in A or B or both. The union is designated as A + B or A ª B, as shown in right half of Figure 3 below.

Figure 3: Intersection and Union of Two Events

2 Axioms of Event Algebra

Using the set abstractions presented above, it is possible to construct a event logic of events or an event algebra. For example if we wish to represent the occurrence of either an event A or an event B or both, we could use the union or sum notation: A + B. On the other hand, if we wish to represent the joint occurrence of both A and B, we could use the product or intersection notation: AB. The following list of seven axioms provides a logical foundation for this event algebra.

A + B = B + A
(A')' = A
A + (B + C) = (A + B) + C
AB = (A' + B')' or (AB)' = A' + B'
A + (BC) = (A + B)(A +C)
AA' = {}
AI = A

3 Theorems of Event Algebra

All of the following theorems can be derived from the seven axioms presented in the previous section.

AB = BA

A(BC) = (AB)C

(A'B')' = A + B or A'B' = (A+B)'

A(B+C) = AB + AC

A + A' = I

A + {}= A

A{}= {}

A + I = I

A + A = A

AA = A

A + AB = A

A + A'B = A + B

4 Mutually Exclusive and Collectively Exhaustive Events

Events in a sample space are mutually exclusive if none of the events intersect one another. In other words, there are no elementary events that are contained in more than one event. Events in a sample space are collectively exhaustive if every elementary element in the sample space is contained in at least one event set. A sample space may consist of events that are both mutually exclusive and collectively exhaustive. The Venn diagrams in Figure 4 illustrate these definitions.

Figure 4: Mutual Exclusivity and Collective Exhaustiveness

Although two events may not be mutually exclusive, we can always represent some combination of these events in mutually exclusive form. For example, the union of each of the following mutually exclusive event sets is equal to the set A + B.

Figure 5: Choices in Mutually Exclusive Form

5 Exercises

(1) Prove the theorems of event algebra using only the seven axioms given in section 2 and substitutions or name changes.

For example, prove theorem 6:

Prove A + {}= A

A I = A , Axiom 7

A' + I' = A' , Axiom 4

A' + {}= A' , I'={}, definition

A + {}= A , Change names

Another example, prove Theorem 3:

Prove (A'B')' = A + B

(AB)' = A' + B' , Axiom 4

(A'B')' = A + B , Change names

(2) Represent the following concepts in event algebra and Venn diagram form.

(a) Event A is a necessary condition for event B.

(b) Event A is a sufficient condition for event B

(3) Represent the following event sets (A1, A2, A3, and A4) based on the four given elementary events (E1, E2, E3, and E4) in Venn diagram form. Assume that events E1 and E4 are mutually exclusive.

Elementary Events

E1: drink champagne

E2: have a party

E3: go to class

E4: stay sober

Event Sets

A1: drink champagne and go to class

A2: have a party and go to class

A3: stay sober and go to class

A4: drink champagne, have a party, and go to class

6 References

R. Howard, Mathematics Associated with Systems Engineering, Chapter 38, "Probability", pp. 3-47, Cambridge, MA: MIT Press.

Siddall, James, N. Probabilistic Engineering Design: Principles and Applications, New York: Marcel Dekker, Inc., 1983, pp. 13-18.

ME290MExpert Systems in Mechanical Engineering