Question

The Financial News Daily has 25 reporters covering Asia, 20 covering Europe, and 20 covering North America. Four reporters cover Asia and Europe but not North America, 6 reporters cover Asia and North America but not Europe, and 7 reporters cover Europe and North America but not Asia. How many reporters cover all of the 3 continents (Asia, Europe, and North America)? (1) The Financial News Daily has 38 reporters in total covering at least 1 of the following continents: Asia, Europe, and North America. (2) There are more Financial News Daily reporters covering only Asia than there are Financial News Daily reporters covering only North America.

Short answer

There are 4 reporters covering all three continents.

Step-by-step solution

- Define Variables

Let A be the number of reporters covering Asia, E be the number covering Europe, and N be the number covering North America. Let x be the number of reporters covering all three continents.

- Given Numbers

The problem states: A = 25, E = 20, N = 20. Additionally, there are 4 reporters covering Asia and Europe but not North America, 6 covering Asia and North America but not Europe, and 7 covering Europe and North America but not Asia.

- Set Up the Inclusion-Exclusion Principle

According to the principle: \[ |A \cup E \cup N| = |A| + |E| + |N| - |A \cap E| - |A \cap N| - |E \cap N| + |A \cap E \cap N| \]

- Account for Intersections

From the problem: \[ |A \cap E| = 4 + x \]\[ |A \cap N| = 6 + x \]\[ |E \cap N| = 7 + x \]

- Insert Values into the Inclusion-Exclusion Formula

Substitute all values including intersections into the inclusion-exclusion formula: \[ 38 = 25 + 20 + 20 - (4 + x) - (6 + x) - (7 + x) + x \]

- Solve the Equation

Simplify and solve: \[ 38 = 65 - 17 \] which simplifies to: \[ 38 - x = 48 - x \] resulting in: \[ x = 4 \]

Key concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a powerful tool in set theory and probability. It's especially useful when dealing with overlapping sets.
In essence, this principle helps you find the union of multiple sets by adjusting for overlap. If you just add the sizes of the sets, you overcount the elements that appear in more than one set.
The formula looks like this:
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \] The terms subtracted represent the overlaps of the sets taken two at a time, and the term added back represents the overlap of all three sets.
This tool is very powerful for solving problems involving unions and intersections of multiple sets.

Set Intersections

Intersections help us understand the commonality between sets.
When we talk about the intersection of two sets like A and E, we denote it as \(|A \cap E|\). This represents elements that are in both sets A and E.
Here's how it works in our problem:

• \(||A \cap E|| = 4 + x\), where 4 reporters cover both Asia and Europe but not North America, and x represents those who cover all three continents.
• Similarly, \(||A \cap N|| = 6 + x\) and \(||E \cap N|| = 7 + x\).

This intersection method provides a granular view of the overlaps and helps in applying the Inclusion-Exclusion Principle effectively.

Logical Reasoning

Logical reasoning is crucial for problem-solving. It helps you break down complex problems into manageable parts.
In our example, understand the logical flow of the problem:

• We define variables representing the number of reporters covering each continent and intersections.
• Next, we account for given and hidden overlaps.
• We then set up the problem using the Inclusion-Exclusion Principle and solve it step-by-step.

By following this logical path, you ensure all parts of the problem are addressed.
This logical flow enables you to navigate through the intricacies of the problem methodically.

Problem-Solving in Mathematics

Effective problem-solving involves several steps:
1. Clearly understand the problem. Read it multiple times if needed.
2. Identify all given information and what is being asked.
3. Define variables for unknown quantities.
4. Break down the problem using fundamental principles like set theory or the Inclusion-Exclusion Principle.
5. Perform step-by-step calculations.
6. Double-check your work to ensure accuracy.
In our example, by following these steps, we derived the solution for the number of reporters covering all three continents:
We first organized our data and then used the Inclusion-Exclusion Principle to set up an equation
Solving it step by step gave us the correct count for those covering all continents.
This structured approach not only makes the problem solvable but also enables you to apply similar steps to different mathematical problems.