Problem 1 - Discrete Mathematics with Applications

Question

A survey conducted among 300 adult shows that 160 like to have their houses painted white and 140 like blue. Seventy-four like both colors. How many do not like either color?

Step-by-Step Solution

Answer

There are 74 adults who do not like either white or blue colors for their houses.

Step 1: Define the variables

Let's define the variables for this problem: - Let A represent the set of adults who like their houses painted white. - Let B represent the set of adults who like their houses painted blue. - There are |A| = 160 adults who like white. - There are |B| = 140 adults who like blue. - There are |A ∩ B| = 74 adults who like both white and blue. - Total adults = 300

Step 2: Apply the principle of inclusion-exclusion

The principle of inclusion-exclusion states that the total number of adults who like at least one of the colors (white or blue) is given by: |A ∪ B| = |A| + |B| - |A ∩ B| Substitute the given values into the formula: |A ∪ B| = 160 + 140 - 74 = 226 So, there are 226 adults who like at least one of the colors.

Step 3: Find the number of adults who do not like either color

Now, we will find the number of adults who do not like either color, i.e., neither white nor blue. To do this, we subtract the number of adults who like at least one of the colors from the total number of adults: Number of adults who do not like either color = Total adults - |A ∪ B| = 300 - 226 = 74

Step 4: Conclusion

There are 74 adults who do not like either white or blue colors for their houses.

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with the collection of objects, considered as a set. In our problem, we can discuss sets using very simple terms— think of them as groups of things that share a common attribute. For example, in this survey problem, we have two sets: one consisting of adults who like their houses painted white, and another consisting of those who prefer blue.

In set theory, when we talk about a set, we often use notation like \( A \) for the set of white-liking adults and \( B \) for blue-liking adults. The quantity or size of a set is denoted by \(|A|\) or \(|B|\), which tells us how many elements (or adults, in this case) are in that set. For this exercise, \(|A| = 160\) and \(|B| = 140\).

Set theory also allows us to understand relationships between groups. For instance, the intersection of two sets, \( A \cap B \), helps us find elements common to both sets—in our case, adults who like both colors. This is crucial for organizing and solving survey-related questions effectively.

Survey Analysis

Survey analysis involves examining responses to identify patterns or trends. In an example like ours, it means figuring out not just who likes each color, but who likes both, and even who doesn’t like either.

In this problem, we analyze a survey of 300 adults to discover their house color preferences. Through proper interpretation techniques, we determine:

• 160 adults like white (\(|A| = 160\)).
• 140 adults like blue (\(|B| = 140\)).
• 74 adults like both blue and white (\(|A \cap B| = 74\)).

To uncover interesting insights, such as the number of people who do not prefer these colors at all, the key lies in accurately handling and analyzing the given data.

Problem Solving Steps

Effective problem-solving often involves a step-by-step process to arrive at a solution. The principle of inclusion-exclusion is vital here. It helps address the overlap between sets—those liking both colors—so we don't double-count them.

Applying the Principle of Inclusion-Exclusion

This principle tells us how to combine quantities from overlapping sets:

• First, characterize the sets and their intersections.
• Express the union of sets with \(|A \cup B| = |A| + |B| - |A \cap B|\).

After applying the formula, we find that 226 people like at least one color.

Finding the Total for "Neither"

To determine how many prefer neither color, subtract the number who like at least one from the total surveyed:

• Total minus the union: \(300 - 226 = 74\).

This organized step-by-step approach ensures that the problem is systematically resolved, leading to an accurate conclusion.