Intersection and Union of Sets: A Mathematical Exploration in Everyday Preferences

The interplay between intersection and union of sets can be intriguing, especially when applied to everyday scenarios. This article delves into a specific problem involving the preferences of a group of individuals for tea, milk, and both, using set theory to solve the puzzle. Understanding these concepts can help in various real-world applications, from marketing to data analysis.

Understanding the Problem

A recent survey reported interesting data on the preferences of a group of people for tea and milk. Here's what the data showed:

One-fifth of people like tea only. 16 people did not like tea at all. 60 people like milk, but 8 of those did not like tea at all.

Solving the Problem Using Set Theory

To resolve this problem, we can use the principles of set theory. Let's break it down step by step.

First Pass

We start by defining the variables:

Let A represent the set of people who like tea. Let B represent the set of people who like milk. Let AB represent the set of people who like both tea and milk. Let A union B represent the total number of people who like either tea or milk or both. Let A intersection B represent the set of people who like both tea and milk. Let A not B represent the set of people who like tea but not milk. Let B not A represent the set of people who like milk but not tea.

From the given data:

Like tea only: 1/5 of the total group. Like milk only: 16 - 8 8 (since 8 people do not like tea at all). Like both: To be determined. Like none: 8 people.

Let's denote the total number of people as T. We need to find the number of people who like both tea and milk.

Second Pass

Using the information from the first pass, let's recompile the data:

Like tea only: 20 people (since 1/5 of the total group). This means 20 out of T people like tea but not milk. Like milk only and like both: 60 people. This means 60 out of T people like milk but not tea. Like none: 8 people.

Now, let's find the total number of people who like either tea or milk or both:

Total people Like tea only Like milk only Like both Like none

Let x be the number of people who like both tea and milk. Then:

T 20 8 x 8

Since the total number of people is the sum of all subsets, we can also express it as:

T (Like tea only Like both) (Like milk only Like both) Like none

T (20 x) (8 x) 8

By simplifying:

20 8 x 8 8 (20 x) (8 x) 36 x 24 2x 12 x

Therefore, the number of people who like both tea and milk is 12.

Conclusion

The principles of set theory can be incredibly powerful tools in solving problems involving overlapping sets. In this case, we used sets to determine how many people like both tea and milk based on the given data. This method can be applied to other scenarios, such as analyzing customer preferences or market research data.

Related Keywords

Set theory Intersection Union Mathematical logic

Additional Resources

Introduction to Set Theory Union and Intersection of Sets Calculator for Intersection and Union