Mathematical Methods for Physics and Engineering - Matematica.NET

30.1 VENN DIAGRAMSA42 6 3BS15Figure 30.2 The Venn diagram for the outcomes of the die-throwing trialsdescribed in the worked example.ABABS(a)S(b)ABAĀS(c)S(d)Figure 30.3 Venn diagrams: the shaded regions show (a) A ∩ B, theintersectionof two events A and B, (b)A ∪ B, the union of events A and B, (c)the complement Ā of an event A, (d)A − B, those outcomes in A that do notbelong to B.graphically explicit the fact that A and B are disjoint. It is not necessary, however,to draw the diagram in this way, since we may simply assign zero outcomes tothe shaded region in figure 30.3(a). An event that contains no outcomes is calledthe empty event and denoted by ∅. The event comprising all the elements thatbelong to either A or B, or to both, is called the union of A and B and is denotedby A ∪ B (see figure 30.3(b)). In the previous example, A ∪ B = {2, 3, 4, 6}.It is sometimes convenient to talk about those outcomes that do not belong toa particular event. The set of outcomes that do not belong to A is called thecomplement of A and is denoted by Ā (see figure 30.3(c)); this can also be writtenas Ā = S − A. It is clear that A ∪ Ā = S and A ∩ Ā = ∅.The above notation can be extended in an obvious way, so that A − B denotesthe outcomes in A that do not belong to B. It is clear from figure 30.3(d) thatA − B canalsobewrittenasA ∩ ¯B. Finally, when all the outcomes in event B(say) also belong to event A, but A may contain, in addition, outcomes that do1121