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Chapter 0: Problem 22

In Exercises \(21-28,\) find the intersection of the sets. $$(1,3,7) \cap(2,3,8)$$

Short Answer

Expert verified
The intersection of the sets (1,3,7) and (2,3,8) is (3).

Step by step solution

01

Identify the sets

Identify the two sets that are provided in the exercise, which are (1,3,7) and (2,3,8).
02

Compare the elements of the sets

The next step is to compare the elements in each set to determine which numbers are present in both. We can see that both sets contain the number 3, so that is the intersection of our two sets.
03

Write down the intersection

Having found the common element(s), we can now write the intersection of the two sets. The intersection is the set that contains only the common elements of the both sets. In this case the intersection would be (3).

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Key Concepts

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

Sets
Sets are a foundational concept in mathematics, often used to group objects, numbers, or elements. A set is usually denoted with curly brackets, for example, \( \{1, 3, 7\} \). Each object in a set is called an element. An important property of sets is that they are unordered collections, meaning the arrangement of elements does not matter. For instance, \( \{1, 3, 7\} \) is considered the same set as \( \{7, 3, 1\} \).

Sets can hold any kind of object, but in most problems, we typically deal with numbers or simple objects.
  • A set with no elements is called an "empty set" and is denoted by \( \emptyset \) or \( \{ \} \).
  • In set notation, repetition of an element is ignored, so \( \{1, 1, 3, 7\} \) is effectively the same as \( \{1, 3, 7\} \).
Understanding sets is essential as they are a basis for many mathematical operations and concepts, including intersections, unions, and subsets.
Common Elements
Common elements refer to the items that appear in each of two or more given sets. These are the elements that sets share. In simple terms, if you have two sets, you find the common elements by checking which elements appear in both sets.

For example, given two sets \( \{1, 3, 7\} \) and \( \{2, 3, 8\} \), the common element is 3 because it is the only number that appears in both sets.
  • Finding common elements is similar to looking for similarities between two lists.
  • Common elements are essential in many applications, from database searches to simplifying equations in algebra.
Understanding how to identify common elements helps you better grasp problems involving set operations, such as intersections.
Comparing Sets
Comparing sets involves looking at the elements they contain and determining their relationships. This could mean checking for common elements, as in intersections, or looking for differences and similarities. Comparing sets is crucial for understanding how sets interact with each other.

There are several ways to compare sets:
  • Intersection: The intersection of two sets is a new set containing all common elements. For example, the intersection of \( \{1, 3, 7\} \) and \( \{2, 3, 8\} \) is \( \{3\} \).
  • Union: This operation combines all unique elements from both sets, eliminating duplicates.
  • Difference: The difference between two sets includes elements present in one set but not in the other.
Knowing how to compare sets and use these operations allows you to analyze and solve problems that involve various set interactions.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I factored \(4 x^{2}-100\) completely and obtained \((2 x+10)(2 x-10)\).

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In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and \(2012 .\) Also shown is the percentage of households in which a person of faith is married to someone with no religion. GRAPH CAN'T COPY. The formula $$I=\frac{1}{4} x+26$$ models the percentage of U.S. households with an interfaith marriage, \(I, x\) years after \(1988 .\) The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, \(N, x\) years after \(1988 .\) Use these models to solve Exercises \(107-108\). a. In which years will more than \(34 \%\) of U.S. households have an interfaith marriage? b. In which years will more than \(15 \%\) of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than \(34 \%\) of households have an interfaith marriage and more than \(15 \%\) have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than \(34 \%\) of households have an interfaith marriage or more than \(15 \%\) have a faith/no religion marriage?

Explain how to factor the difference of two squares. Provide an example with your explanation.

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