Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Find the events \(E^{c}\) and \(F^{c} \cap G\).

Step-by-Step Solution

Answer

The events \(E^{c}\) and \(F^{c} \cap G\) can be found as follows: 1. Find the complement of E (E^c): \(E^{c} = \{c, d, e, f\}\) 2. Find the complement of F (F^c): \(F^{c} = \{b, c, e\}\) 3. Find the intersection of F^c and G (F^c ∩ G): \(F^{c} \cap G = \{b, c, e\}\)

Step 1: Find the complement of E (E^c)

To find E^c, we need to find all elements in the sample space S that are not in E. Since E = {a, b}, all other elements in S that are not a or b are in the complement of E. E^c = {c, d, e, f}

Step 2: Find the complement of F (F^c)

To find F^c, we need to find all elements in the sample space S that are not in F. Since F = {a, d, f}, all other elements in S that are not a, d, or f are in the complement of F. F^c = {b, c, e}

Step 3: Find the intersection of F^c and G (F^c ∩ G)

To find F^c ∩ G, we need to find all elements that are in both F^c and G. F^c = {b, c, e} G = {b, c, e} The common elements in F^c and G are b, c, e. Therefore, F^c ∩ G = {b, c, e}

Key Concepts

Sample Space

Understanding the concept of a sample space is crucial when dealing with probability and set theory. A sample space is essentially the set of all possible outcomes of a particular experiment or situation. It is denoted by the letter \(S\). For example, in the given exercise, the sample space \(S\) is \(\{a, b, c, d, e, f\}\). This means that these are all the possible outcomes that can occur in the experiment.
Knowing the sample space helps you see the full scope of possibilities and is the foundation for defining events, which are subsets of the sample space. Without a clear understanding of the sample space, it's challenging to understand probabilities or analyze particular events.

Events

Events are specific outcomes or groups of outcomes from the sample space that we are interested in. In set theory, each event is a subset of the sample space. For instance, in the exercise \(E = \{a, b\}\), \(F = \{a, d, f\}\), and \(G = \{b, c, e\}\) are all events that occur based on the defined sample space \(S\).
Consider an event as a scenario or a specific occurrence that we are analyzing or measuring. Events can also be combined using operations such as union, intersection, and complement to develop more complex questions and solutions.

Complement of a Set

The complement of a set, denoted as \(A^c\), comprises all the elements in the sample space \(S\) that are not in the set \(A\). It's like looking at what's missing from the set, relative to the sample space. For example, for the set \(E = \{a, b\}\), its complement, \(E^c\), is all elements in \(S\) that are not in \(E\). So \(E^c = \{c, d, e, f\}\).
Computing complements is an essential operation in set theory, which helps us understand what an event doesn't include, adding another layer to our analysis of probabilities and possibilities.

Intersection of Sets

The intersection of two sets refers to the elements that are common to both sets. In mathematical terms, this is represented by the symbol \(\cap\). When you find \(A \cap B\), you are looking for elements that both sets have in common. In our exercise, we determine \(F^c \cap G\) by identifying elements present in both \(F^c\) and \(G\).
For \(F^c = \{b, c, e\}\) and \(G = \{b, c, e\}\), their intersection is straightforward, as they share all elements: \(F^c \cap G = \{b, c, e\}\). This operation is useful when we need to explore shared properties or outcomes between sets, which is frequently needed in probability calculations and logical operations.