Question

If \(\mathrm{A}=\\{1,2,3,4\\}\) then which of the following are functions from A to itself? \((1) f_{1}=\\{(x, y): y=x+1\\}\) (2) \(f_{2}=\\{(x, y): x+y>4\\}\) (3) \(\mathrm{f}_{3}=\\{(\mathrm{x}, \mathrm{y}): \mathrm{x}<\mathrm{y}\\}\) (4) \(\mathrm{f}_{4}=\\{(\mathrm{x}, \mathrm{y}): \mathrm{x}+\mathrm{y}=5\\}\)

Short answer

Only \(f_4\) is a function.

Step-by-step solution

- Function Definition

A function from set A to itself, denoted as \(f: A \to A\), must satisfy the condition that every element x in set A is paired with exactly one element y in set A.

- Evaluate \(f_1 = \{(x, y): y = x + 1\}\)

For each element in A (which are 1, 2, 3, 4), check if the resulting y (where y = x + 1) is in A. For x = 1, y = 2 (in A). For x = 2, y = 3 (in A). For x = 3, y = 4 (in A). For x = 4, y = 5 (not in A). Not all elements of A have a y in A, thus \(f_1\) is not a function.

- Evaluate \(f_2 = \{(x, y): x + y > 4\}\)

Check each element in A to see if it forms a valid pair with an element in A. For x = 1, possible y-values are 4. For x = 2, possible y-values are 3, 4. For x = 3, possible y-values are 2, 3, 4. For x = 4, possible y-values are 1, 2, 3, 4. Multiple y-values exist for each x, so this does not satisfy the function definition. Thus, \(f_2\) is not a function.

- Evaluate \(f_3 = \{(x, y): x < y\}\)

Check each element in A if it satisfies the condition. For x = 1, possible y-values are 2, 3, 4. For x = 2, possible y-values are 3, 4. For x = 3, possible y-values are 4. For x = 4, no valid y-values exist. As all x values must have exactly one y-value, this mapping does not satisfy a function. Thus, \(f_3\) is not a function.

- Evaluate \(f_4 = \{(x, y): x + y = 5\}\)

Check each element in A to see if one unique element in A pairs according to the given condition. For x = 1, y = 4 (in A). For x = 2, y = 3 (in A). For x = 3, y = 2 (in A). For x = 4, y = 1 (in A). Each x has one and only one corresponding y in A, thus \(f_4\) is a function.

Key concepts

Mathematical Functions

A mathematical function is a special kind of relation between two sets. It assigns exactly one element of the second set to each element of the first set. Functions can be represented in different ways:

• As an equation, e.g., \( f(x) = x^2 \)
• Through a graph, showing the relationship between elements
• By specifying pairs of values (x, y)

If we have a set A, a function from A to itself means every element in A is mapped to another element in A. For instance, in our exercise with set A = {1, 2, 3, 4}, we want to see if each x in A has a corresponding y in A. The following conditions must be met for a relation to be a function from A to A: each element of A must map to one and only one element of A.

Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. In our problem, the set A = {1, 2, 3, 4} consists of four numbers. Important concepts in set theory include:

• Elements: The objects in a set, e.g., 1, 2, 3, and 4 in set A.
• Subsets: A set containing some or all elements of another set, e.g., {1, 2} is a subset of A.
• Set Notation: Defined using curly braces, e.g., A = {1, 2, 3, 4}.

When dealing with functions from a set to itself, as in our exercise, we need to ensure that the functions align with the elements and subsets of A. Understanding sets helps grasp how functions operate within specific boundaries.

Function Definition

Defining a function is crucial to understanding how it works. Here are the key points to remember:

• A function f from set A to set B is denoted as \( f: A \rightarrow B \).
• Each element x in A is paired with exactly one element y in B.
• The pair (x, y) must satisfy the function's rule.

Let's look at our solutions:

• \( f_1 \) : \( y = x + 1 \). This pair fails because 4 maps to 5, which is not in A.
• \( f_2 \) : \( x + y > 4 \). This pair fails as x has multiple y-values.
• \( f_3 \) : \( x < y \). Fails because not every x finds a valid y in A.
• \( f_4 \) : \( x + y = 5 \). This pair works because each x maps to exactly one y in A.

By following these steps, we ensure that a relation qualifies as a function when it meets all criteria accurately.