Problem 11 - Discrete Mathematics with Applications

Question

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lceil x+y\rceil$$

Step-by-Step Solution

Answer

\(x = 3.456\) and \(y = 2.789\) \(x + y = 6.245\) \(\lceil x + y \rceil = \lceil 6.245 \rceil = 7\)

Step 1: Calculate the sum of x and y

Add the given numbers \(x = 3.456\) and \(y = 2.789\) together to get their sum: $$ x + y = 3.456 + 2.789 $$

Step 2: Compute the sum

Calculate the result of the sum from the previous step: $$ x + y = 6.245 $$

Step 3: Apply the ceiling function

Now, apply the ceiling function to the sum, which will give us the smallest integer greater than or equal to the sum: $$ \lceil x + y \rceil = \lceil 6.245 \rceil $$

Step 4: Find the result

Finally, determine the smallest integer that is greater than or equal to the sum: $$ \lceil x + y \rceil = 7 $$ So, the result is \(\lceil x+y\rceil = 7\).

Key Concepts

Discrete Mathematics and the Ceiling Function

Discrete mathematics is a branch of mathematics that deals with discrete elements that use distinct values. Unlike continuous mathematics, where data can take on any value within a range, discrete mathematics works with numbers that have clear gaps between them. An example of a discrete function in mathematics is the ceiling function, which is an essential part of number theory and combinatorics, both fields within discrete mathematics.

When dealing with the ceiling function, represented by the symbol \( \lceil \cdot \rceil \), you're typically tasked with identifying the smallest integer that is greater than or equal to a given real number. This operation transforms a real number into an integer by rounding up. In the problem where you have two real numbers, \(x = 3.456\) and \(y = 2.789\), after adding them together, the ceiling function helps in finding the integer part of the resulting sum without considering its decimal part. This makes it a discrete operation, as it results in an integer value from a set of distinct options.

Mathematical Operations: Adding Real Numbers and More

Mathematical operations are procedures or functions applied to one or more values (operands) to attain a result. In the context of this exercise, adding two real numbers is the initial operation. A real number is a value that represents a quantity along a continuous line, which can include both decimal and integer values. The sum of two real numbers is also a real number.

To deepen your understanding, let's further dissect the process as follows:

Addition of Real Numbers

Starting with the given real numbers \(x = 3.456\) and \(y = 2.789\), addition combines these quantities into a single sum \(x + y = 6.245\), which is still a real number with a decimal part.

Application of the Ceiling Function

Post addition, when we apply the ceiling function to the sum, mathematical operations transition from producing a real number to yielding an integer. This mapping to the nearest integer is what provides a discrete result from continuous data.

Real Numbers in Mathematical Problems

Real numbers include the entire spectrum of numbers used in mathematics, which comprises both rational and irrational numbers. This means that any number we can find on the number line, including decimals, fractions, and integers, belongs to the set of real numbers.

In our exercise, both \(x = 3.456\) and \(y = 2.789\) are real numbers and they represent quantities that can be mapped on the number line with precision. Their addition yields another real number \(x + y = 6.245\). However, when we apply the ceiling function, we convert this real number into an integer, indicating a key difference between continuous and discrete values.

Understanding Real Numbers

In practical terms, real numbers can be used to measure continuous quantities like distance or time, but when it comes to counting distinct objects or units, integers from the real numbers set are commonly used. This split within real numbers—between continuous decimal values and whole integers—lays the foundation for various mathematical operations, highlighting the versatility and depth of real numbers in both theoretical and applied mathematics.