Understanding the Commuter Set Theory Problem

The problem describes a group of 50 people and their modes of transport to work: train, bus, or neither. We are given the number of people who use only train (implicitly, as part of the set of train users), only bus (implicitly, as part of the set of bus users), and neither. We need to find the number of people who use both train and bus.

This is a classic problem that can be solved using the principles of set theory, specifically the concept of union and intersection of sets, often visualized with a Venn diagram.

Applying Set Theory for Transport Analysis

Let's define the sets:

• Let \(U\) be the universal set of all 50 persons. So, \(|U| = 50\).
• Let \(T\) be the set of persons who go to work by train. We are given \(|T| = 18\).
• Let \(B\) be the set of persons who go to work by bus. We are given \(|B| = 26\).
• Let \(N\) be the set of persons who go by neither train nor bus. We are given \(|N| = 8\).

The set of persons who go by either train or bus or both is represented by the union of sets \(T\) and \(B\), denoted as \(T \cup B\). The persons who use neither train nor bus are outside this union.

The total number of people is the sum of those who use train or bus (or both) and those who use neither:

\[|U| = |T \cup B| + |N|\]

We know \(|U| = 50\) and \(|N| = 8\). We can find the number of people who use either train or bus or both:

\[|T \cup B| = |U| - |N|\]

\[|T \cup B| = 50 - 8\]

\[|T \cup B| = 42\]

So, 42 people use either train or bus or both.

Calculating the Intersection (Both Train and Bus Users)

We need to find the number of people who use both train and bus. This is the intersection of sets \(T\) and \(B\), denoted as \(|T \cap B|\).

The principle of inclusion-exclusion for two sets states:

\[|T \cup B| = |T| + |B| - |T \cap B|\]

We know \(|T \cup B| = 42\), \(|T| = 18\), and \(|B| = 26\). We can substitute these values into the formula:

\[42 = 18 + 26 - |T \cap B|\]

First, add \(|T|\) and \(|B|\):

\[18 + 26 = 44\]

So the equation becomes:

\[42 = 44 - |T \cap B|\]

To find \(|T \cap B|\), rearrange the equation:

\[|T \cap B| = 44 - 42\]

\[|T \cap B| = 2\]

Therefore, 2 people use both train and bus to go to work.

Summarizing the Commuter Data

We can summarize the results:

The calculation shows that 2 people use both modes of transport.

Revision Table: Key Concepts for Set Theory Problems

Additional Information on Solving Set Theory Problems

Set theory problems like this one are common in aptitude tests and competitive exams. They help assess logical reasoning and understanding of basic set operations. Here are some additional points:

• Venn Diagrams: Drawing a Venn diagram is a very helpful way to visualize these problems. Draw two overlapping circles (representing Train and Bus) inside a rectangle (representing the Total group). The overlapping area is the intersection (Both), the parts of the circles outside the overlap are the 'only' users (Train Only, Bus Only), and the area outside the circles but inside the rectangle is the 'neither' group.
• 'Only' vs 'Total' in a Set: Be careful to distinguish between people who use *only* train and people who use train (which includes those who use both train and bus). In this problem, the given numbers 18 and 26 represent the total number of people in the Train set and the Bus set, respectively, including those in the intersection.
• Generalization: The principle of inclusion-exclusion can be extended to three or more sets. For three sets A, B, and C: \(|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|\).

By carefully applying the inclusion-exclusion principle and understanding the relationship between the total group, those using a specific mode, and those using neither, we can accurately determine the number of individuals using both modes of transport.