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Chapter 6: Problem 32

Web Searches In January 2009 , a search using the Web search engine GoogleTs for "foreclosure" yielded 55 million Web sites containing that word. A search for "meltdown" yielded 17 million sites. A search for sites containing both words yielded 1 million sites. \({ }^{4}\) How many Web sites contained either "foreclosure" or "meltdown" or both? HIMT [See Example 1.]

Short Answer

Expert verified
There were 71 million websites containing either "foreclosure" or "meltdown" or both in January 2009, using the addition principle in set theory.

Step by step solution

01

Identify the given information

We are given the following information: - Number of websites containing the word "foreclosure": 55 million - Number of websites containing the word "meltdown": 17 million - Number of websites containing both words: 1 million
02

Use the addition principle

According to the addition principle in set theory, to find the number of websites containing either "foreclosure" or "meltdown" or both, we need to add the number of websites containing "foreclosure" and the number of websites containing "meltdown", and then subtract the number of websites containing both words to avoid double counting. Mathematically, this can be represented as: Number of websites containing either word = (Number of websites containing "foreclosure") + (Number of websites containing "meltdown") - (Number of websites containing both words)
03

Calculate the number of websites containing either word

Plug in the given values into the formula: Number of websites containing either word = (55 million) + (17 million) - (1 million) Now, perform the calculations: Number of websites containing either word = 71 million Therefore, 71 million websites contained either "foreclosure" or "meltdown" or both in January 2009.

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Key Concepts

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

Addition Principle
The addition principle is a fundamental concept in set theory that helps us count the number of elements in the union of different sets. If you have two sets, A and B, and want to know how many elements are in either set, you follow a specific process. Here鈥檚 how it works:
  • Add the number of elements in set A.
  • Add the number of elements in set B.
  • Subtract the number of elements that are in both sets A and B (the intersection) since these elements are counted twice.
This gives us the formula:\[|A \cup B| = |A| + |B| - |A \cap B|\]In our exercise, this principle helps determine the number of websites that mention either "foreclosure," "meltdown," or both. By applying the addition principle, we discover the total count without duplicating the ones that feature both terms.
Problem Solving
Problem-solving within the context of set theory often involves breaking down complex information into manageable steps. Let's see how this is applied in our exercise:
  • Step 1: Identify and Understand the Given Information
    Start by clearly outlining what data you have. In this instance, the problem provides the number of websites for each keyword as well as for both.
  • Step 2: Choose and Apply the Relevant Concept
    Then, apply a suitable mathematical concept. Here, it's the addition principle from set theory.
  • Step 3: Execute Calculations
    Next, calculate using the values given and the formula derived from your concept to find the solution.
  • Step 4: Interpret the Result
    Finally, reflect on the result to ensure it makes sense in the context of the problem.
Breaking the problem down in this way clarifies the process and ensures each part is addressed effectively, leading to accurate conclusions.
Web Searches
Understanding how web searches and keywords work is crucial for interpreting and solving problems similar to our exercise. Search engines like Google index vast amounts of information from various websites, allowing users to find pages containing particular words.
  • Indexes
    Search engines store indexed information about web pages, categorizing them based on keywords.
  • Keyword Searches
    A search query targets these keywords to find relevant pages. For example, searching for "foreclosure" will pull all indexed pages containing that word.
  • Boolean Logic
    Combining keywords (using terms like "and," "or") utilizes Boolean logic, refining search results further. Our exercise uses this logic implicitly when considering websites with either or both keywords.
Understanding web search mechanisms is essential, especially when using mathematical principles like the addition principle, ensuring you interpret the data and context correctly and efficiently.

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