Problem 37 - Discrete Mathematics with Applications

Question

Let \(S=\\{\text { true, falsel. Define a boolean function } f : \mathbb{N} \rightarrow S \text { by } f(n)=\text { true }\) if year \(n\) is a leap year and false otherwise. Find \(f(n)\) for each year \(n .\) $$2076$$

Step-by-Step Solution

Answer

For the given year \(n=2076\), we know that it is divisible by 4, but not divisible by 100, which means it is a leap year. Therefore, the function value for the boolean function \(f(n)\) is $$f(2076) = \text{true}.$$

Step 1: Recall the rules for leap years

To classify a particular year as a leap year, it needs to meet one or more given conditions: 1. The year must be divisible by 4. 2. If the year is divisible by 100, it must also be divisible by 400. We will apply these rules to the year 2076 to determine if it is a leap year.

Step 2: Check if the year is divisible by 4

To know if the year is divisible by 4, we check if the remainder of the year divided by 4 is equal to 0. In this case, \(2076 \div 4 = 519\), with no remainder. Since 2076 is evenly divisible by 4, we can proceed to the next step.

Step 3: Check if the year is divisible by 100

If a year is divisible by 100, we need to check if it is also divisible by 400 to determine if it's a leap year. 2076 is not divisible by 100, so we don't need to check the second condition. In this case, it's already clear that 2076 is a leap year.

Step 4: Determine the function value for the given year

Since we have concluded that 2076 is a leap year, we can now find the function value of \(f(n)\) for \(n=2076\). According to the definition, \(f(n)=\text{true}\) if year \(n\) is a leap year. Therefore, $$f(2076) = \text{true}.$$

Key Concepts

Leap Year

A leap year is a special year in the calendar that contains one extra day, making it 366 days long instead of the regular 365 days. This additional day is added to the month of February, giving it 29 days instead of 28. Leap years occur due to the Earth's orbit around the Sun, which takes approximately 365.25 days. To account for the extra quarter day each year, a full day is added every four years.

Here's how you can determine if a year is a leap year:

• The year must be divisible by 4.
• If the year is divisible by 100, it must also be divisible by 400 to be a leap year.

For instance, the year 2076 can be checked using these rules. First, 2076 divided by 4 leaves no remainder, so it satisfies the first condition. Since 2076 is not divisible by 100, there is no need to check the second condition. Hence, 2076 is indeed a leap year.

Natural Numbers

Natural numbers are the basic set of numbers used for counting and ordering. These numbers begin from 1 and continue infinitely. They are simple and used frequently in everyday math, from counting objects to numbering in sequences.

Natural numbers include:

• Positive integers starting from 1: 1, 2, 3, 4, 5, and so forth.
• They do not include negative numbers, decimals, or fractions.

They are often represented using the symbol \(\mathbb{N}\). In the exercise, when considering the year 2076 as a natural number, it aligns perfectly because all years are natural numbers due to their positive, whole number status.

Divisibility Rules

Divisibility rules help us quickly determine whether a number can be divided by another without leaving a remainder. These shortcuts are particularly handy when assessing larger numbers, as they save time and simplify calculations.

Key divisibility rules include:

• Divisible by 2: A number must be even, which means it ends in 0, 2, 4, 6, or 8.
• Divisible by 3: The sum of the number's digits must be divisible by 3.
• Divisible by 4: Only the last two digits need to form a number divided evenly by 4.
• Divisible by 5: Numbers must end in 0 or 5.
• Divisible by 6: The number must meet the criteria for divisibility by both 2 and 3.

In our example, to check if 2076 is a leap year, the divisibility rule for 4 was applied. Since 2076 is evenly divisible by 4, it meets one of the conditions necessary for identifying it as a leap year.