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Venn diagrams

closed curves representing the events A and B do not overlap, so as to make 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 to the shaded region in Fig. 1.3. An event that contains
no outcomes is called the empty event and denoted by ∅. The event comprising all the elements
that belong to either A or B, or to both, is called theunion of A and B and is denoted by A∪B (see
Fig. 1.4). In the previous example, A ∪ B = {2, 3, 4, 6}. It is sometimes convenient to talk about
those outcomes that do not belong to a particular event. The set of outcomes that do not belong
¯
to A is called the complement of A and is denoted by A (see Fig. 1.5); this can also be written as
¯
¯
¯
A = S − A. It is clear that A ∩ A = S and A ∩ A = ∅.

A B

S

Figure 1.3 – The shaded region show A ∩ B, the intersection of two events A and B

A B

S
Figure 1.4 – The shaded region show A ∪ B, the union of two events A and B

A

A

S
¯
Figure 1.5 – The shaded region show A, the complement of an event A

The above notation can be extended in an obvious way, so that A−B denotes the outcomes in
¯
A that do not belong to B. It is clear from Fig. 1.6 that A − B can also be written as A ∩ B. Finally,

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