Event Relations and Operations

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In the last section, we have defined events as subsets of the sample space . In interpreting events as sets, we can apply the same operations and relations to events, that we know from basic set theory. We shall now recapitulate some of the most important concepts of set theory.

Subsets and Complements

is subset of is denoted by   Thus if event occurs, occurs as well. and are equivalent events if and only if (abbreviated as ’iff’) and . If then we define the complement of  denoted by to be the set of points in that are not in ,

Union of Sets

The set of points belonging to either the set or the set is called the union of sets and , and is denoted by .  Thus if the event ’’ has occurred, then a basic outcome in the set has taken place.

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Set union can be extended to sets and hence events : in which case we have Example: Rolling a die once Define and Then General results: where is the sample space. where is the null set, the set with no elements in it.

Intersection of Sets

The set of points common to the sets AND is known as intersection of and , .  Thus if the event ’’ has occurred, then a basic outcome in the set has taken place.

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Set intersection can be extended to sets and hence to events : Example: Rolling a die once Define and Then General results: Disjoint events: Two sets or events are to said to be disjoint (or mutually exclusive) if their intersection is the empty set: . Interpretation: events and cannot occur simultaneously. By definition, and are mutually exclusive. The reverse doesn’t hold, i.e. disjoint events aren’t necessarily complements of each other. Example: Rolling a die once Define and Then and Interpretation: events (sets) and are disjoint and complementary events. Define and Interpretation: events and are disjoint but not complementary.

Logical Difference of Sets or Events

The set or event is the logical difference of events and if it represents the event: ’ has occurred but has not occurred’ i.e. it is the outcomes in , that are not in :

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Example: Rolling a six-sided die once Define and Then and

Disjoint Decomposition of the Sample Space

A set of events is called disjoint decomposition of , if the following conditions hold:

One can think of such a decomposition as a partition of the sample space where each basic outcome falls into exactly one set or event. Sharing a birthday cake results in a disjoint decomposition or partition of the cake. Example: Rolling a six-sided dice Sample space: Define . Claim: one possible disjoint decomposition is given by . Proof: ,,,,,,.

Some Set Theoretic Laws

De Morgan’s laws Associative laws Commutative laws Distributive laws

Summary

Verbal Technical Algebraic
If occurs, then occurs also is subset of
and always occur together and are equivalent events
and cannot occur together and are disjoint events
occurs if and only does not occur and are complementary events
occurs if and only if at least one occurs is union of
occurs if and only if all occur is intersection of all