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Chapter 3: Problem 5

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x+y\rfloor$$

Short Answer

Expert verified
The value of \(\lfloor x+y\rfloor\) for the given values of \(x\) and \(y\) is 6.

Step by step solution

01

Add the given numbers

Add the given numbers \(x\) and \(y\) as follows: $$ x+y = 3.456 + 2.789 $$
02

Simplify the sum

Now, calculate the sum of this addition: $$ x+y = 3.456 + 2.789 = 6.245 $$
03

Apply the floor function

Now that we have the sum of \(x\) and \(y\), we will find the greatest integer less than or equal to this sum by applying the floor function: $$ \lfloor x+y\rfloor = \lfloor 6.245\rfloor $$
04

Evaluate the floor function

Finally, determine the greatest integer less than or equal to the calculated sum: $$ \lfloor x+y\rfloor = \lfloor 6.245\rfloor = 6 $$ The value of \(\lfloor x+y\rfloor\) for the given values of \(x\) and \(y\) is 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Floor Function
The floor function, represented by \( \lfloor x \rfloor \), is a mathematical concept that deals with the rounding down of real numbers. Essentially, the floor function refers to finding the greatest integer that is less than or equal to a given number. For instance, if the number is 6.245, applying the floor function will yield 6. This is because 6 is the largest integer not greater than 6.245.

Key characteristics of the floor function include:
  • It can handle both positive and negative numbers. For example, \( \lfloor -2.3 \rfloor = -3 \) as \(-3\) is less than \(-2.3\) but still an integer.
  • It always rounds towards negative infinity, meaning it's not about typical rounding rules.
  • It is different from the ceiling function, which rounds up to the nearest integer.
Understanding this function is crucial in discrete mathematics as it helps in decision-making scenarios, such as computing maxima or simplifying expressions.
Addition
Addition is one of the most basic operations in arithmetic and mathematics, involving the calculation of the total of two or more numbers or amounts. The numbers added together are called "addends", and the result is the "sum" of the addition. For the problem of adding \(x = 3.456\) and \(y = 2.789\), the operation is performed as follows:

  • Add the decimal parts together: 0.456 + 0.789 = 1.245. Here, carry over the 1 to the whole number part.
  • Add the whole number parts together, including the carry from the decimal part: 3 + 2 + 1 = 6.
  • The sum is thus \(6.245\).
When performing addition, pay attention to aligning decimal points for accuracy. This practice ensures that each digit is added to its corresponding place value.
Greatest Integer Function
The greatest integer function is often synonymous with the floor function in many contexts, and it represents the same idea of rounding down to the nearest integer. The notation \( \lfloor x \rfloor \) conveys this.
  • A useful property of the greatest integer function is \( x - 1 < \lfloor x \rfloor \leq x \). This means the greatest integer is always less than or equal to \(x\) but absolutely closest to it below.
  • For the expression \( \lfloor 6.245 \rfloor \), it simplifies directly to 6, as 6 is the nearest integer below 6.245.
  • In contrast, if you deal with negative values, it gets more interesting: \( \lfloor -1.5 \rfloor = -2 \).
The greatest integer function is a vital part of functions involving division and modular arithmetic, making it a tool of great importance in discrete math.

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Most popular questions from this chapter

Prove. The open interval \((a, b)\) is uncountable. [Hint: Find a suitable bijection from \((0,1)\) to \((a, b) . ]\)

Prove. The set of irrational numbers is uncountable. (Hint: Prove by contradiction.)

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{j=1}^{6} \sum_{i=1}^{5}\left(i^{2}-j+1\right) $$

Evaluate each sum and product, where \(p\) is a prime and \(I=\\{1,2,3,5\\}.\) $$\begin{aligned} &\prod(3 i-1)\\\ &i \in I \end{aligned}$$

Prove. The cartesian product of two countable sets is countable.
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