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Chapter 7: Problem 38

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 \(A \subseteq B\), then \(n(B)=n(A)+n\left(A^{c} \cap B\right)\).

Short Answer

Expert verified
The statement is true. If \(A \subseteq B\), then the number of elements in B equals the sum of the number of elements in A and the number of elements in the intersection of the complement of A and B, or \(n(B) = n(A) + n(A^{c} \cap B)\). This is because B can be considered the union of two disjoint sets A and \(A^{c} \cap B\), for which the cardinality of the union is equal to the sum of their cardinalities.

Step by step solution

01

Understand the terms used in the statement

Try to understand each symbol and term used in the statement: - \(A \subseteq B\) means that A is a subset of B (i.e., all elements of A are also elements of B). - \(n(X)\) indicates the cardinality or the number of elements in the set X. - \(A^{c}\) represents the complement of the set A (i.e., all elements not in A). - \(A^{c} \cap B\) means the intersection of the complement of A and set B (i.e., all elements that belong to both the complement of A and B).
02

Rewrite the statement in a logical order

Rephrase the statement to understand the relation between the terms: "If A is a subset of B, then the number of elements in B equals the sum of the number of elements in A and the number of elements in the intersection of the complement of A and B."
03

Apply set theory concepts

Now, recall the properties of sets and utilize them to prove the statement. Consider B as the union of the two disjoint sets A and \(A^{c} \cap B\), because there are no elements that belong to both A and \(A^{c} \cap B\). \(B = A \cup (A^{c} \cap B)\) For disjoint sets, the cardinality of the union is equal to the sum of their cardinalities: \(n(B) = n(A) + n(A^{c} \cap B)\)
04

Conclude the result

Since we were able to demonstrate that for any A being a subset of B, the cardinality of B is equal to the sum of the cardinalities of A and the intersection of the complement of A with B, we can conclude that the given statement is true.

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Most popular questions from this chapter

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. One die shows a 6 , and the other is a number less than 3 .

Let \(S=\left\\{s_{1}, s_{2}, s_{3}, s_{4}\right\\}\) be the sample space associated with an experiment having the probability distribution shown in the accompanying table. If \(A=\left\\{s_{1}, s_{2}\right\\}\) and \(B=\left\\{s_{1}, s_{3}\right\\}\), find a. \(P(A), P(B)\) b. \(P\left(A^{c}\right), P\left(B^{c}\right)\) c. \(P(A \cap B)\) d. \(P(A \cup B)\) $$ \begin{array}{lc} \hline \text { Outcome } & \text { Probability } \\ \hline s_{1} & \frac{1}{8} \\ \hline s_{2} & \frac{3}{8} \\ \hline s_{3} & \frac{1}{4} \\ \hline s_{4} & \frac{1}{4} \\ \hline \end{array} $$

STAYING IN ToucH In a poll conducted in 2007, 2000 adults ages 18 yr and older were asked how frequently they are in touch with their parents by phone. The results of the poll are as follows: $$ \begin{array}{lccccc} \hline \text { Answer } & \text { Monthly } & \text { Weekly } & \text { Daily } & \text { Don't know } & \text { Less } \\ \hline \text { Respondents, \% } & 11 & 47 & 32 & 2 & 8 \\ \hline \end{array} $$ If a person who participated in the poll is selected at random, what is the probability that the person said he or she kept in touch with his or her parents a. Once a week? b. At least once a week?

List the simple events associated with each experiment. An opinion poll is conducted among a group of registered voters. Their political affiliationDemocrat (D), Republican ( \(R\) ), or Independent \((I)\) -and their sex-male \((m)\) or female \((f)-\) are recorded.

The manager of a local bank observes how long it takes a customer to complete his transactions at the automatic bank teller. a. Describe an appropriate sample space for this experiment. b. Describe the event that it takes a customer between 2 and 3 min to complete his transactions at the automatic bank teller.
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