There are four main set operations which include set union, set intersection, set complement, and set disjoint. In this section, we will learn the various set operations, notations of representing sets, how to operate on sets, and their usage in real life.
1. Union (A∪B):
The union of two events A and B is the set of all elements that are in A, in B, or in both. In terms of probability,
P(A∪B) = P(A)+P(B) − P(A∩B).
Example: Suppose we have a deck of 52 cards. Let event A be drawing a red card, and event B be drawing a face card (Jack, Queen, King). There are 26 red cards and 12 face cards (6 red & 6 black).
- P(A) = 26/52 = 0.5
- P(B) = 12/52 ≈ 0.231
- P(A∩B) = 6/52 ≈ 0.115
P(A∪B) = P(A)+P(B) − P(A∩B) = 0.5 + 0.231 − 0.115 ≈ 0.616
2. Intersection (A∩B):
The intersection of two events A and B is the set of elements that are in both A and B. In terms of probability, P(A∩B) is the probability that both A and B occur.
P(A∩B) = P(A).P(B)
Example: Consider rolling two dice. Let event A be rolling a 4 on the first die, and event B be rolling an even number on the second die.
- P(A) = 1/6 ≈ 0.16
- P(B) = 3/6 = 0.5
P(A∩B)=P(A).P(B) = 0.16⋅0.5 ≈ 0.083
3. Complement (Ac):
The complement of an event A is the set of all outcomes in the sample space that are not in A. In terms of probability, P(Ac)=1−P(A).
Example: If the probability of raining today (event A) is 0.3, then the probability of not raining today (event Ac) is:
P(Ac)=1−P(A) = 1 − 0.3 = 0.7
4. Disjoint (Mutually Exclusive) Events:
Two events A and B are mutually exclusive if they cannot both occur at the same time, i.e., A∩B=∅. If A and B are disjoint, then P(A∪B)=P(A)+P(B).
Example: Consider drawing a card from a deck. Let event A be drawing a heart, and event BB be drawing a club. These two events are mutually exclusive as one card cannot be both a heart and a club.
- P(A) = 13/52 = 0.25
- P(B) = 13/52 = 0.25
Since they are disjoint, P(A∪B) = P(A)+P(B) = 0.25+0.25 = 0.5
