Problem 2 - Applied Discrete Structures

Question

Find the ranges of the following functions on \(\mathbb{Z}\) : (a) \(g=\\{(x, 4 x+1) \mid x \in \mathbb{Z}\\}\). (b) \(h(x)=\) the least integer that is greater than or equal to \(\sqrt{|x|}\). (c) \(P(x)=x+10\).

Step-by-Step Solution

Answer

(a) Range is \(4k+1 : k \in \mathbb{Z}\); (b) Range is all positive integers; (c) Range is \(\mathbb{Z}\).

Step 1: Analyze Function g(x)

The function given is defined as tuples \(g = \{(x, 4x + 1) \mid x \in \mathbb{Z}\}\). This implies that for each integer \(x\), there is an output \(4x + 1\). Since \(4x\) takes every integer value that is a multiple of 4, and adding 1 shifts these values by 1, the range consists of every integer of the form \(4k + 1\) where \(k\) is an integer.

Step 2: Determine Function h(x) Range

Function \(h(x)\) returns the ceiling value of \(\sqrt{|x|}\). For each non-negative integer \(x\), compute \(\sqrt{x}\) and take the ceiling value (least integer greater or equal to this value). For instance, \(\sqrt{0} = 0\), \(\sqrt{1} = 1\), and \(\sqrt{3} \approx 1.732\) where ceiling value is 2. The range consists of all positive integers since as \(x\) increases, all higher integers can be attained as least integer values.

Step 3: Analyze Function P(x)

The function is given by \(P(x) = x + 10\). For each integer \(x\), the output is an integer \(x\) increased by 10. Therefore, the range of \(P(x)\) is all integers \(y\) where \(y = x + 10\) and \(x\) is an integer, resulting in all integers \(..., -8, -7, ..., 9, 10, 11, ...\). The range is \(\mathbb{Z}\), the set of all integers.

Key Concepts

Function Range

The range of a function is essentially the set of all possible outputs a function can produce. When working with functions, especially in discrete mathematics, understanding the range is crucial. For the set of functions on the integer domain \( \mathbb{Z} \), the range is determined by how the function rules map integers to outputs.

• For the function \( g = \{(x, 4x + 1) \mid x \in \mathbb{Z}\} \): Each input integer \( x \) is transformed into an output of the form \( 4x + 1 \). This shows that the output is every integer that can be expressed as \( 4k + 1 \), where \( k \) is any integer.
• The range of \( P(x) = x + 10 \) is all integers because you are simply adding 10 to every integer \( x \). This operation does not exclude any integer from being an output, implying the complete set of integers, denoted by \( \mathbb{Z} \), is the range.
• For functions with a variable exponent or radical operation, like \( h(x) = \lceil \sqrt{|x|} \rceil \), analyzing the range involves understanding the particular properties of the involved mathematical operations. Here, the range is all positive integers as you map each non-negative integer \( x \) through the ceiling function of its square root.

Ceiling Function

The ceiling function is a valuable tool in discrete mathematics. It rounds up any real number to the least integer greater than or equal to it. Understanding how it operates on a set of values, especially within functions, reveals insights into possible outputs or ranges.

• The operation \( h(x) = \lceil \sqrt{|x|} \rceil \) takes the absolute value of \( x \), finds its square root, and then applies the ceiling function. This operation effectively maps a non-negative integer to the smallest integer not less than the square root of \( x \).
• For instance, when \( x = 3 \), \( \sqrt{3} \approx 1.732 \), and the ceiling of \( 1.732 \) is 2. These details help delineate a function's range when employing the ceiling function.
• Given its property of always rounding up, the ceiling function often results in outputs that are integers greater than or equal to any fractional input resulting from other operations (e.g., square roots, divisions, etc.).

Integer Functions

Integer functions are functions uniquely tied to integer inputs and outputs, characterized by their operations constrained within the integer set. This distinction from real or rational numbers allows for discrete analysis and understanding.

• When analyzing functions such as \( P(x) = x + 10 \), we're focusing exclusively on integer additions, ensuring that the output remains within the integer domain, \( \mathbb{Z} \). The resulting function maps any integer \( x \) to another integer \( y \), producing no gaps or omissions in the output set.
• The function \( g = \{(x, 4x + 1) \mid x \in \mathbb{Z}\} \) is a classic example of an integer function where outputs are specifically reduced to a subset of integers, following the general form \( 4k + 1 \).
• Integer functions can be utilized to model countable events, sequences, or scenarios where fractional values are inherently impractical or undesirable, using entirely whole number operations.