Set Notation for Events

Various combinations of the occurrence of ai two events A and B can be indicated in set notation as follows  

A -I- B A occurs or B occurs in the mutually inclusive sense to indicate the occurrence of A, B, or botli A and B. 

Venn diagrams (Fig. 19.3.1) provide a pictorial representation of these events. 

In set terminology A is called tlie complement of A, that is, the set of elements in S that are not in A. Altonate notation for the complement of A is A or A. AB is called Uie interseaion of A and B, that is, tlie set of elements in both A and B. Alternate notation in the literature for tlie intersection of A and B is A 

When events A and B liave no elements in common tliey are said to be mutually exclusive. A set liaving no elements is called tlie null set and is designated by i. Tlius if events A and B are mutually exclusive then AB = i. 

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After defining fundamental terms used in probability and introducing set notation for events, we consider probability theorems facilitating tlie calculation of the probabilities of complex events. Conditional probability and tlie concept of independence lead to Bayes theorem and tlie means it provides for revision of probabilities on tlie basis of additional evidence. Random variables, llicir probability distributions, and expected values provide tlie means... 

Shared phenomena may be events, operation calls, messages, and the like. They are observable by at least two domains, but controlled by only one domain, as indicated by an exclamation mark. For example, in Fig. 1 the notation 0 E4 means that the phenomena in the set E4 are controlled by the domain Operator. 

Here, we follow the notation introduced in Chap. 7. The reader is referred to Sect.7.3.1 for more details. It will be useful to recall sets 7, Lt, and AHf hr It is also assumed that there are L events in which uncertainty ( ) unfolds over the planning horizon. The value that random factors can take in event / can be identified by index hi which belongs to set Hi. Then, ft/ is the combination of event realizations identified by (ft i, /t2, , ft )- 7 is the subset of planning periods t that are associated with uncertain event 1. Lt is the reciprocal set of T/. Lt is the event / which is related... 

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