Problem 35 - 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 \(n\). $$1996$$

Step-by-Step Solution

Answer

The given year is 1996. Since 1996 is divisible by 4 and is not divisible by 100, it is considered a leap year. Thus, the boolean function of 1996 is: \[ f(1996) = \text{true} \]

Step 1: Check for divisibility by 4

To determine if the year 1996 is divisible by 4, we can simply divide it by 4 and check for a remainder. If the remainder is 0, then the condition is met. \[ 1996 \div 4 = 499 \] The result is an integer (499) with no remainder, which means the first condition is met.

Step 2: Check for divisibility by 100

Since leap years divisible by 100 must also be divisible by 400, we need to check if 1996 is divisible by 100. If it is, then we proceed to check divisibility by 400. However, if it isn't, there's no need to check the 400 rule. \[ 1996 \div 100 = 19.96 \] The result is not an integer, which means 1996 is not divisible by 100. Therefore, we don't need to check for divisibility by 400.

Step 3: Determine if 1996 is a leap year

Since 1996 is divisible by 4 and is not divisible by 100, it is considered a leap year according to the leap year rules. Thus, the boolean function of 1996 is: \[ f(1996) = \text{true} \]

Key Concepts

Leap year calculation

Calculating if a year is a leap year requires the application of specific divisibility rules. A year is called a leap year if it meets certain conditions. Primarily, a year must be divisible by 4. This means if you divide the year by 4, there should be no remainder. For instance, dividing 1996 by 4 yields 499 with no remainder, fulfilling this criterion.

However, there's a special rule for years that end in 00, like 1900 or 2000. These years need to be checked for divisibility by 400 to be leap years. If a year is divisible by both 100 and not by 400, it is not a leap year. For example, 1900 is divisible by 100 but not by 400; hence, it's not a leap year. In contrast, 2000 satisfies both conditions, making it a leap year.

As a result, while 1996 satisfies the condition of being divisible by 4 and is not subject to the rule of 100, because it doesn't end with 00, it is considered a leap year.

Divisibility rules

Understanding divisibility rules is crucial for leap year determination and other mathematical problems. These rules simplify the process of determining whether one number can be divided by another without a remainder, ensuring calculations are swift and accurate.

For instance, to check if a number is divisible by 4, simply look at the last two digits of the number. If they form a number divisible by 4, then the whole number is divisible by 4. With 1996, the last two digits "96" are divisible by 4, confirming its divisibility.

Another key rule is for the number 100. A number is divisible by 100 if its last two digits are zeroes. Divisibility by 400 combines these rules; if a number is divisible by both 100 and 4, it's divisible by 400. Remembering these concise rules helps determine divisibility easily and effectively.

Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This means it focuses on distinct and separate values, often involving integers and logical statements, unlike calculus which engages with the continuity of real numbers.

A primary application of discrete mathematics is in computer science, as it underpins binary systems used in programming and logic gates. Boolean functions, like the one determining a leap year, are a classic example. Here, the function outputs either "true" or "false", aligning with discrete outputs.

In the leap year problem, discrete math aids in crafting logical boolean functions to deliver decisions about social calendars based on simple yes/no or true/false determinations. By understanding these underpinnings, students gain insight into the structured processes that define mathematical, computer, and decision-making sciences.