Construct a Venn diagram portraying four events A, B, C, and D such that the collection of events A, B, and C is mutually exclusive, the collection of events A, B, and D is mutually exclusive, but the collection of events A, B, C, and D is not mutually exclusive.

Question

Quizplus · Accepted Answer

The Answer is To construct a Venn diagram that satisfies the given conditions, we need to understand the requirements:

1. The collection of events A, B, and C is mutually exclusive, which means that these events cannot occur at the same time. In other words, there is no overlap between A, B, and C in the Venn diagram.
2. The collection of events A, B, and D is mutually exclusive, which similarly means that A, B, and D cannot occur at the same time, and there is no overlap between them in the Venn diagram.
3. The collection of events A, B, C, and D is not mutually exclusive, which means that there is some overlap among these four events.

Given these conditions, we can infer that the overlap must involve event D with either C or both. Since A, B, and C are mutually exclusive, and A, B, and D are mutually exclusive, D cannot overlap with A or B. Therefore, D must overlap with C.

Here's how to draw the Venn diagram:

1. Draw three non-overlapping circles and label them A, B, and C. This represents the mutual exclusivity of A, B, and C.
2. Draw a fourth circle for event D. This circle should not overlap with A or B, but it should overlap with C to represent that D can occur with C, but not with A or B.

The resulting Venn diagram will show four circles where A, B, and C are separate, and D overlaps only with C. This satisfies the condition that A, B, and C as well as A, B, and D are mutually exclusive, but A, B, C, and D are not mutually exclusive because C and D can occur together.

Here's a textual representation of the Venn diagram:

```
A B
( ) ( )
C D
( ) ( )
```

In this diagram, the circles for A and B are separate, the circle for C is separate from A and B, and the circle for D overlaps with C but not with A or B.

Construct a Venn Diagram Portraying Four Events A, B, C