Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a subset of \(B\) and \(P(B)=0\), then \(P(A)=0\).

Step-by-Step Solution

Answer

The statement is true. If \(A\) is a subset of \(B\) and \(P(B) = 0\), then \(P(A) = 0\) because the condition \(P(A) \le P(B)\) is satisfied, and probabilities cannot be negative.

Step 1: Recall the definition of a subset

A set \(A\) is a subset of a set \(B\) if every element of \(A\) is also an element of \(B\). In other words, \(A \subseteq B\) means that if \(x \in A\), then \(x \in B\).

Step 2: Recall the properties of probabilities

Probabilities are values that range from 0 to 1, i.e., \(0 \le P(E) \le 1\) where E represents any event. Additionally, if an event is a subset of another event, the probability of the first event cannot be greater than the probability of the second event. In other words, if \(A \subseteq B\), then \(P(A) \le P(B)\).

Step 3: Apply the properties of probabilities to the given statement

Now we can apply these properties to the given statement. If \(A\) is a subset of \(B\) and \(P(B) = 0\), then, since probabilities cannot be negative, the only possible value for \(P(A)\) that satisfies the condition \(P(A) \le P(B)\) is \(P(A) = 0\).

Step 4: Conclusion

Therefore, the statement is true. If \(A\) is a subset of \(B\) and \(P(B) = 0\), then \(P(A) = 0\).

Key Concepts

Understanding Subsets

The idea of a subset is a fundamental concept in set theory and plays a crucial role in probability theory. When we say that a set \(A\) is a subset of another set \(B\), it means that every element in \(A\) is also in \(B\). We can express this relationship using the symbol \(A \subseteq B\). Think of it like a basket within a larger basket; everything in the smaller basket can also be found in the larger one.

Here are a few points to make it clearer:

• If \(x \in A\), then it must follow that \(x \in B\).
• Even if \(A\) has no elements (i.e., it is an empty set), it is still considered a subset of \(B\).
• This relationship is central in the fields of mathematics and helps to establish the characteristics shared between different sets.

By understanding that \(A\) is contained within \(B\), we can start exploring how this affects probabilities and other mathematical properties.

Exploring Probability Properties

Probabilities measure how likely an event is to occur and range from 0 to 1. These numbers provide a clear boundary; \(0\) indicates impossibility, while \(1\) indicates certainty. When it comes to subsets in probability, there are some important properties to remember.

The key property is that for any events \(A\) and \(B\), if \(A\) is a subset of \(B\) (\(A \subseteq B\)), the probability of \(A\) cannot exceed the probability of \(B\). Mathematically, this is expressed as:

• \(P(A) \le P(B)\)

If the probability of \(B\) is \(0\), meaning \(B\) cannot occur, any subset of \(B\), must also have a probability of \(0\). Therefore, \(P(A) = 0\) when \(P(B) = 0\), ensuring consistency in probability calculations and rules.

Basic Elements of Set Theory

Set theory provides the language and principles for discussing collections of objects, referred to as sets. It is the foundation upon which many areas of mathematics are built, including probability theory.

At its core, set theory deals with the classification and relationships of groups of objects. Here's a quick rundown of some essential points:

• A **set** is a collection of distinct objects, considered as an object in its own right.
• Sets can include anything: numbers, symbols, or even other sets, with no repeated elements.
• The **empty set**, denoted \(\emptyset\), is a vital concept which represents a set with no elements.
• Understanding relationships like union, intersection, and subset helps structure complex mathematical ideas.

These basic concepts empower the study of probability by establishing a framework to define and analyze events, such as when determining subset relationships and calculating the likelihood of events.