Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S=\\{1,2,3,4,5,6\\}, E=\\{2,4,6\\}\) \(F=\\{1,3,5\\}\), and \(G=\\{5,6\\}\). Are the events \(F\) and \(G\) mutually exclusive?

Step-by-Step Solution

Answer

No, the events F and G are not mutually exclusive, as their intersection \(F∩G=\\{5\\}\) is not an empty set. They have a common outcome (5), which means they can both occur at the same time.

Step 1: Understand the concept of mutually exclusive events

Two events A and B are called mutually exclusive if their intersection is the empty set, which means they have no common outcomes. In other words, \(A∩B= Ø\).

Step 2: Determine the intersection of F and G

To check if F and G are mutually exclusive, we need to find their intersection. The intersection of two sets is a set containing elements that are common to both sets. So, we will find the intersection of F and G: \(F = \\{1,3,5\\}\) \(G = \\{5,6\\}\) \(F∩G=\\{5\\}\)

Step 3: Check if F and G are mutually exclusive

Since the intersection of F and G is not an empty set (\(F∩G=\\{5\\}\)), the events F and G are not mutually exclusive. They have a common outcome (5), which means they can both occur at the same time.

Key Concepts

Intersection of Sets

When discussing the fundamentals of set theory, one often starts with the concept of an 'intersection' of sets. The intersection refers to a new set formed by all the elements that are common to two or more given sets.

Let's consider two simple sets, A and B. The intersection of A and B, denoted by A ∩ B, is the set containing all the elements that are both in A and in B. If there are no common elements, then the intersection is the empty set, often represented by Ø or {}.

Applying this to the problem at hand, where we have sets F and G with elements \(F=\{1,3,5\}\) and \(G=\{5,6\}\), the intersection, denoted by \(F∩G\), would be the set that contains all numbers that are present in both F and G. As identified in the problem solution, F and G share the number 5, which means \(F∩G=\{5\}\). This common element indicates that F and G do intersect and are therefore not mutually exclusive.

Probability Theory

Moving into the broader realm of probability theory, let's explore its connection with the concept of mutually exclusive events. In probability, two events are deemed mutually exclusive if they cannot occur at the same time. This is analogous to the intersection of sets being empty—there are no overlapping outcomes between the events.

For example, when flipping a fair coin, the events 'landing on heads' and 'landing on tails' are mutually exclusive. You can't have both outcomes on a single coin flip. In mathematical terms, this translates to these events having no intersection, which aligns with the definition in set theory.

In the context of our specific exercise from the textbook, the sets F and G represent possible outcomes of some event, like rolling a dice. Since their intersection is not empty (they share the number 5), they can occur simultaneously in whatever scenario they're representing—perhaps picking a number from a bag. Therefore, in probability terms, F and G are not mutually exclusive events.

Set Theory

Lastly, let's dig a bit deeper into set theory, a mathematical science that studies sets, which are fundamentally collections of objects. These objects could be anything: numbers, symbols, points in space, etc. Set theory forms the basis for many areas of mathematics, including parts of probability.

Within set theory, we develop ways to combine, relate, and analyze collections of objects through operations such as union, intersection, and difference. Union combines all distinct elements of two sets, while difference includes only elements present in one set but not the other. The power of set theory is evident as it applies to many fields within and outside of mathematics, offering a structured way to handle grouped data or situations.

In summary, every concept in the exercise involving the sets S, E, F, and G relates back to these fundamental principles of set theory. Understanding set theory not only helps resolve the problem at hand but also builds a foundation for more complex mathematical and logical reasoning.