Problem 38 - 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 \(n .\) $$3000$$

Step-by-Step Solution

Answer

The given year is 3000. We check if it is a leap year by following the conditions: divisible by 4, not divisible by 100, or divisible by 400. 3000 is divisible by 4 and 100, but not by 400. Therefore, it does not qualify as a leap year. Thus, the function \(f(3000)\) returns \(\text{false}\).

Step 1: Determine if the number is divisible by 4

To figure out if the given number (3000) is divisible by 4, divide 3000 by 4 and check if the result is a whole number: \[ \frac{3000}{4} = 750 \] Since 750 is a whole number, 3000 is divisible by 4.

Step 2: Determine if the number is divisible by 100

Next, determine if the given number (3000) is divisible by 100. Divide 3000 by 100 and check if the result is a whole number: \[ \frac{3000}{100} = 30 \] Since 30 is a whole number, 3000 is divisible by 100.

Step 3: Determine if the number is divisible by 400

Lastly, determine if the given number (3000) is divisible by 400. Divide 3000 by 400 and check if the result is a whole number: \[ \frac{3000}{400} = 7.5 \] Since 7.5 is not a whole number, 3000 is not divisible by 400.

Step 4: Apply the leap year conditions

If a year is divisible by 4 and not divisible by 100, or if it is divisible by both 100 and 400, it is a leap year. In this case, 3000 is divisible by 4 and by 100, but not divisible by 400. According to the leap year conditions, it is not a leap year. Thus, the function f(3000) returns false, because 3000 is not a leap year.

Key Concepts

Leap Year

A leap year is a special year in our Gregorian calendar that allocates an extra day, making it 366 days long instead of the usual 365 days. This adjustment is necessary to keep our calendar in alignment with Earth's revolutions around the Sun. Otherwise, our calendar would gradually drift over time.

• A year is identified as a leap year if it is divisible by 4.
• If the year can also be evenly divided by 100, it must be further divisible by 400 to be a leap year.

For example, the year 2000 was a leap year because it is divisible by 4, 100, and 400. On the other hand, the year 1900, though divisible by 4 and 100, was not a leap year because it is not divisible by 400.
So, in the context of the exercise, the year 3000 follows the leap year rules; it’s divisible by 4 and 100 but not by 400, thus not a leap year.

Divisibility

Divisibility is a mathematical concept used to determine if one integer can be exactly divided by another, with no remainder left. It's a useful principle that helps simplify problems, like determining leap years or working with other number properties.
Understanding whether a number is divisible by another involves the following steps:

• Divide the number and check that the result is a whole number (integer).
• If the division results in a fraction or decimal, the number is not divisible by the divisor.
• The concept of divisibility is often used to establish conditions, like those for leap years - divisibility by 4, 100, and sometimes 400.

For instance, dividing 3000 by 4 gives us 750, meaning it's divisible by 4. However, when dividing by 400, the result is 7.5, not a whole number, indicating 3000 is not divisible by 400.

Conditional Statements

Conditional statements are logical statements used to make decisions based on certain conditions being met. In mathematics, these statements often help us set criteria and rules for determining outcomes, like whether a year is a leap year.
Conditional statements typically take the format of "if... then... else..." clauses, organizing logic structures in programming or mathematical scenarios.

• The basic form is: If condition A is true, then do action B.
• If the condition is false, do some other action C.
• This logic helps in testing various scenarios and applying rules, like checking leap year conditions with divisibility tests.

In the leap year exercise, the conditional statement checks multiple conditions:
"If a year is divisible by 4, and either not divisible by 100 or divisible by 400, it is a leap year; otherwise, it is not." In our example of the year 3000, because it does not meet all the conditions, the logical statement returns false, thus determining it is not a leap year.