Problem 36 - Discrete Mathematics with Applications

Question

Let \(S=\\{\text { true, false }\\} .\) Define a boolean function \(f: \mathbb{N} \rightarrow S\) by \(f(n)=\) true if year \(n\) is a leap year and false otherwise. Find \(f(n)\) for each year \(n\). $$2020$$

Step-by-Step Solution

Answer

\(f(2020)=\) true, as 2020 is divisible by 4 and not divisible by 100, satisfying the leap year rules.

Step 1: Check Divisibility by 4

Let's check if 2020 is divisible by 4. To do this, we can divide 2020 by 4 and see if the result is a whole number. \[ 2020\div 4= 505 \] Since 505 is a whole number, 2020 is divisible by 4.

Step 2: Check Divisibility by 100

Now, let's check if 2020 is divisible by 100. To do this, we can divide 2020 by 100. \[ 2020\div 100= 20.20 \] Since 20.20 is not a whole number, the year 2020 is not divisible by 100.

Step 3: Conclusion

Since 2020 is divisible by 4 and not divisible by 100, it satisfies the rules for leap years. Therefore, \(f(2020)=\) true, and the year 2020 is a leap year.

Key Concepts

Leap year

We often hear that a leap year occurs every four years. But what exactly makes a year a leap year? It's all about aligning our calendar year with the solar year. A solar year, which is the time taken for Earth to orbit the Sun, is approximately 365.24 days. A typical year has 365 days, which means there’s an extra 0.24 days per year—too small to put on the calendar monthly but big enough to notice after a few years. This is where the leap year correction helps.

Leap years add an extra day, February 29th, to help synchronize the calendar year with the solar year. However, not every year divisible by 4 is a leap year. The rules are:

• If a year is divisible by 4, it could be a leap year.
• If it is divisible by 100, then it is not a leap year.
• Unless, it is divisible by 400, then it actually is a leap year.

These rules were established to correct the inaccuracies that still exist if we just added a leap day every 4 years.

Divisibility rules

Divisibility rules help us quickly determine whether one number is divisible by another without performing a complete division. These are especially useful with the leap year rules.

To check for divisibility by a particular number, use these rules:

• Divisibility by 4: A number is divisible by 4 if the last two digits form a number that's divisible by 4.
• Divisibility by 100: A number is divisible by 100 if it ends in two zeros.
• Divisibility by 400: A number is divisible by 400 if the last two digits form a number that's divisible by 4 and the hundreds plus two zeros are divisible by 4 (i.e., the number is both divisible by 100 and 4).

These rules help us determine if a year is a leap year quickly. For example, in checking 2020:

• 2020 is divisible by 4 as the last two digits, 20, are divisible by 4.
• It’s not divisible by 100, since it does not end in two zeros.

So, 2020 is a leap year!

Mathematical reasoning

Mathematical reasoning is the logical thought process used to arrive at a conclusion. It involves understanding and applying rules systematically. In the context of determining if a year is a leap year, mathematical reasoning involves:

• Recognizing and applying the correct divisibility rules.
• Following the leap year rules sequentially to determine if a year satisfies the conditions for being a leap year.

To solve such problems, approach with clarity:
- First, identify what is being asked or checked—in this case, whether a year like 2020 is a leap year.
- Apply the divisibility rules for 4, 100, and 400.
- Use logical steps to verify if all conditions meet the requirement for a leap year.

By reasoning systematically, you ensure thorough and correct conclusions without missing any important details. This develops a strong foundation for solving various mathematical problems efficiently.