91Ó°ÊÓ

Set theory is a fundamental part of mathematics that deals with collections of objects, known as sets. A set is simply a collection of distinct items where order doesn't matter.
In our problem, we have specific sets such as the sample space \( S = \{1, 2, 3, 4, 5, 6\} \), and the events \( E = \{2, 4, 6\} \), \( F = \{1, 3, 5\} \), and \( G = \{5, 6\} \).
These sets represent collections of possible outcomes or events. In mathematics, operations such as union, intersection, and complement are used to analyze relationships between these sets, helping us to arrive at deeper insights, like determining if sets \( E \) and \( F \) are complementary.

Probability is the study of likelihood and uncertainty. It measures how likely an event is to occur. In the context of events and sample spaces, probability provides a way to quantify the uncertainty of different outcomes.
For any given event \( A \), its probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space \( S \). Hence, \( P(A) = \frac{|A|}{|S|} \).

• For event \( E \), \( P(E) = \frac{3}{6} = \frac{1}{2} \).
• For event \( F \), \( P(F) = \frac{3}{6} = \frac{1}{2} \).

Understanding probability allows us to interpret and predict the chances of events happening, and it is essential when analyzing complementary events.

The sample space in probability and set theory is the set of all possible outcomes. It's the "universe" of outcomes that can occur when conducting an experiment.
In our example, the sample space is \( S = \{1, 2, 3, 4, 5, 6\} \), containing all potential results. Each number represents a different possible outcome.
For events \( E \) and \( F \) to be complementary, they must cover all elements of the sample space without overlapping.

• The union of \( E \) and \( F \) resulted in the full sample space, \( E \cup F = \{1, 2, 3, 4, 5, 6\} = S \).
• The intersection was empty, \( E \cap F = \emptyset \), indicating no overlap.

Thus, \( E \) and \( F \) are complementary because each outcome in \( S \) is covered by either \( E \) or \( F \), but not both.

Finite mathematics encompasses all mathematical subjects that deal with a limited or finite set of elements. It includes topics such as set theory, probability, and their applications in various fields.
Working with finite sets, such as those in our exercise, involves using elements that are countable and limited in number.
Here, we dealt with finite sets representing possible outcomes and events. Events \( E \), \( F \), and the sample space \( S \), all had a limited number of elements illustrating concepts like complement, which are central to understanding finite mathematics.

• This field helps in developing logical thinking and problem-solving skills.
• It is commonly used in computer science, business, and statistics.

By examining finite collections of outcomes and their intersections and unions, we practice key concepts of finite mathematics.