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Chapter 2: Problem 93

Use the formula for the cardinal number of the union of two sets to solve Exercises 93-96. Set \(A\) contains 17 elements, set \(B\) contains 20 elements, and 6 elements are common to sets \(A\) and \(B\). How many elements are in \(A \cup B\) ?

Short Answer

Expert verified
The cardinal number of the union of two sets \(A\) and \(B\), or in other words the number of unique elements in both sets, is 31.

Step by step solution

01

Identifying the Given Information

The number of elements in set \(A\) (denoted as \(|A|\)) is 17, the number of elements in set \(B\) (denoted as \(|B|\)) is 20, and the number of elements common to both sets (denoted as \(|A \cap B|\)) is 6.
02

Applying the Formula

The cardinal number of the union of sets \(A\) and \(B\) is given by the formula \(|A \cup B| = |A| + |B| - |A \cap B|\). Substituting the given values into the formula, we get \(|A \cup B| = 17 + 20 - 6\).
03

Solving the Expression

Perform the arithmetic operation to find the cardinal number of the union of the two sets. The operation gives us \(|A \cup B| = 31\). This means there are 31 elements in \(A \cup B\) when the size of \(A\) is 17, the size of \(B\) is 20, and the size of their intersection is 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cardinal Number
In set theory, a cardinal number is a measure of the "size" of a set. It tells us how many elements a set contains. For example, if set \( A \) has 17 elements, we say the cardinal number of \( A \) is 17, often noted as \(|A| = 17\). This concept is a fundamental part of understanding sets, as it allows us to talk about and compare different sizes of sets. Comparing cardinal numbers is vital when analyzing different sets and determining relationships between them in set theory.
Union of Sets
The union of sets is a fundamental operation in set theory. When you form the union of two sets, you create a new set containing all elements that are in either set, without repeating any elements. This is noted as \( A \cup B \). In our original example, the union of set \( A \) and set \( B \) includes every unique element present in \( A \) or \( B \). Understanding the union helps us see the complete picture of what elements are present when combining two sets. It鈥檚 similar to mixing two collections and observing the total variety of items without duplicates.
Intersection of Sets
The intersection of sets takes a different approach by finding a common ground. It identifies all elements that are present in both sets. The intersection is symbolized as \( A \cap B \). For instance, if 6 elements are common in both sets \( A \) and \( B \), the intersection \( A \cap B \) comprises these elements. This operation highlights similarities or shared attributes between sets. In practical terms, knowing the intersection helps us understand overlap between collections, such as mutual friends in two social circles.
Mathematical Formula
The mathematical formula related to the previous concepts is crucial for calculating various set operations. When determining the cardinal number of the union of two sets, we use the formula:
  • \(|A \cup B| = |A| + |B| - |A \cap B|\)
This formula efficiently calculates the size of the union \( A \cup B \) by adding the sizes of both sets \( A \) and \( B \), and then subtracting the size of their intersection \( A \cap B \). By subtracting the intersection, we ensure that we don't count shared elements twice. Understanding this formula helps solve many problems in set theory, providing a method to handle overlapping data accurately.

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Most popular questions from this chapter

Use the formula for the cardinal number of the union of two sets to solve Exercises 93-96. set \(A\) contains 8 letters and 9 numbers. Set \(B\) contains 7 letters and 10 numbers. Four letters and 3 numbers are common to both sets \(A\) and \(B\). Find the number of elements in set \(A\) or set \(B\).

Use the formula for the cardinal number of the union of two sets to solve Exercises 93-96. Set \(A\) contains 12 numbers and 18 letters. Set \(B\) contains 14 numbers and 10 letters. One number and 6 letters are common to both sets \(A\) and \(B\). Find the number of elements in set \(A\) or set \(B\).

In the August 2005 issue of Consumer Reports, readers suffering from depression reported that alternative treatments were less effective than prescription drugs. Suppose that 550 readers felt better taking prescription drugs, 220 felt better through meditation, and 45 felt better taking St. John's wort. Furthermore, 95 felt better using prescription drugs and meditation, 17 felt better using prescription drugs and St. John's wort, 35 felt better using meditation and St. John's wort, 15 improved using all three treatments, and 150 improved using none of these treatments. (Hypothetical results are partly based on percentages given in Consumer Reports.) a. How many readers suffering from depression were included in the report? Of those included in the report, b. How many felt better using prescription drugs or meditation? c. How many felt better using St. John's wort only? d. How many improved using prescription drugs and meditation, but not St. John's wort? e. How many improved using prescription drugs or St. John's wort, but not meditation? f. How many improved using exactly two of these treatments? g. How many improved using at least one of these treatments?

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