Question

Which of the following sets are functions of \(\mathrm{x}\) ? \(\mathrm{A}=\\{(5,1),(4,2),(4,3),(6,4)\\}\) \(\mathrm{B}=\\{(\mathrm{x}, \mathrm{y})|\mathrm{y}=| \mathrm{x} \mid\\}\) \(\mathrm{C}=\\{(\mathrm{x}, \mathrm{y})|\mathrm{x}=| \mathrm{y} \mid\\} ?\)

Short answer

Only set B is a function of x, as it satisfies the condition that each x value corresponds to a unique y value, with the relationship \(y = |x|\). Sets A and C do not satisfy this condition and are therefore not functions of x.

Step-by-step solution

Set A: Checking for function

For set A, we are given the specific pairs \((5,1), (4,2), (4,3), (6,4)\). We will check if each x value corresponds to a unique y value. (5,1) has x = 5, y = 1 (4,2) has x = 4, y = 2 (4,3) has x = 4, y = 3 (6,4) has x = 6, y = 4 Here, we can see that the x value 4 has two different y values associated with it (2 and 3). Therefore, set A is not a function.

Set B: Checking for function

In set B, we have the condition that for each \((x, y)\), \(y = |x|\). This implies that for every x value, the y value will be the absolute value of x. Since the absolute value of any number is unique, we can conclude that set B is a function of x.

Set C: Checking for function

In set C, we have the condition that for each \((x, y)\), \(x = |y|\). This means that for every y value, the x value will be the absolute value of y. However, this does not guarantee that each x value will have a unique y value associated with it. Let's take an example: Suppose \(y = 2\), then \(x = |2| = 2\) Suppose \(y = -2\), then \(x = |-2| = 2\) In this case, both y = 2 and y = -2 correspond to the same x value, x = 2. Since each x value does not correspond to a unique y value, set C is not a function of x. As a result, only set B is a function of x.

Key concepts

Understanding Sets in Discrete Mathematics

In discrete mathematics, a set is a collection of distinct elements. Sets are fundamental in both mathematics and computer science, providing a framework to group elements together. They help in organizing data and understanding relationships between different entities.
For example, in the context of the exercise, we are dealing with sets of ordered pairs, which represent a relation between two elements: usually denoted as (x, y).
It's crucial to recognize that when we talk about sets, each element is unique and repetition of elements doesn't hold any significance. This property of sets is intrinsically tied to the concept of functions, which rely on the association of unique inputs to outputs.

The Importance of Unique Values in Functions

For a relation to qualify as a function, every input (or x-value) must be linked to exactly one output (or y-value). This is where the idea of unique values becomes critical.
A function can be visualized as a machine: for each input, it produces a single, definite output. If an x-value results in multiple y-values, the relation cannot be termed as a function.

• Set A: Fails as a function because x = 4 leads to two different outputs, y = 2 and y = 3.
• Set B: Functions as intended since each x-value has one corresponding absolute y-value.
• Set C: Fails because different y-values (2 and -2) yield the same x-value, breaking the rule of unique y-values for each x-value.

Concept of Absolute Value

The absolute value of a number is its distance from zero on the number line, denoted by vertical bars, like |x|. It strips away any sign, leaving a non-negative number.
For example, both 3 and -3 have an absolute value of 3 because they are the same distance away from zero in opposite directions. This makes the notion of absolute value particularly handy when defining functions, as it ensures non-negative outputs.

• In Set B, y = |x| ensures each x results in a singular, unique y, maintaining its status as a function.
• In Set C, x = |y| causes an overlap in x-values for different y-values, contradicting the requirements of a function.

Defining Functions in Discrete Mathematics

A function in discrete mathematics is a specific type of relation that pairs each input with a unique output. This distinctive feature distinguishes functions from general relations.
When dealing with functions, one often considers a set of ordered pairs, where no two pairs have the same first element paired with different second elements.
Functions are powerful tools in mathematics due to their predictable nature; they offer consistency and reliability.
To determine if a set is a function, ensure that every x-value corresponds to one, and only one, y-value. This rule eliminates uncertainty, providing a clear, unambiguous mechanism for mapping inputs to outputs.