Question

Decide whether the set of ordered pairs represents a function from \(A\) to \(B\). \(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) Give reasons for your answers. \(\\{(0,-1),(2,2),(1,-2),(3,0),(1,1)\\}\)

Short answer

The set of ordered pairs does not represent a function from \(A\) to \(B\) because the rule of a function, which states that each input from \(A\) should be related to only one output in \(B\), is violated.

Step-by-step solution

Identifying the elements of set A, set B and the ordered pairs

Firstly, note down the sets \(A = \{0,1,2,3\}\), \(B = \{-2,-1,0,1,2\}\), and the ordered pairs \(\{(0,-1),(2,2),(1,-2),(3,0),(1,1)\}\).

Analyzing the ordered pairs

Now, analyze the ordered pairs. These pairs associate every member of \(A\) with a member of \(B\). The pairs are: \(\{(0,-1),(2,2),(1,-2),(3,0),(1,1)\}\). Each first element in the pair corresponds to a member of \(A\) and each second element corresponds to a member of \(B\).

Checking for Function Rule Violation

While conducting this analysis, it is essential to notice whether there's a violation of the function rule, which is that each element of \(A\) should connect to only one element in \(B\). If there’s a pair where the first element is the same, but the second one differs, that’s a violation. Here, the pair \(1,-2\) and \(1,1\) is the violation, because the number 1 from set \(A\) corresponds to -2 and 1 in \(B\).

Key concepts

Ordered Pairs

When we talk about ordered pairs, we refer to a pair of elements written in a specific order, typically denoted as \((x, y)\). This order matters because it signifies a relationship between two elements from two different sets. For example, in an ordered pair \((a, b)\), "a" is related to "b" in the context of the sets from which they originate.

Ordered pairs are fundamental in mathematics for defining relationships like functions. In the given exercise, ordered pairs such as \((0,-1)\) indicate that 0 from set \(A\)is mapped to -1 in set \(B\).Understanding the order is crucial because swapping "a" with "b" creates a different pair, which reflects a different relationship.

When determining if these pairs represent a function, noting the order helps us check if every element of set \(A\)relates to exactly one element in set \(B\).This concept is vital in ensuring that the pairs abide by the rules necessary to form a valid function.

Set Theory

Set theory is a mathematical theory that deals with sets, which are collections of distinct objects. In the context of the exercise, we have two sets: \(A = \{0,1,2,3\}\)and\(B = \{-2,-1,0,1,2\}\).These sets are the domains and codomains of potential functions formed by the listed ordered pairs.

In set theory, functions are defined from one set to another, and each element in the first set (or the domain) is associated with exactly one element in the second set (the codomain). This association is central to understanding how relations are established in mathematics.

The sets \(A\) and \(B\) provide the elements used in the ordered pairs. Understanding these elements as parts of sets helps clarify the constraints on the function—and specifically, whether a function from \(A\) to \(B\)exists without overlap or ambiguity in this mapping.

Function Rule Violation

For a set of ordered pairs to represent a valid function from one set to another, each element in the domain set must pair with exactly one element in the codomain set. This is known as the function rule. It ensures consistent and unambiguous relationships.

In the exercise, the function rule is violated because the element \(1\) in set\(A\)is paired with both \(-2\) and \(1\) in \(B\).This results in conflicting relationships, as an element in the domain set has more than one image in the codomain.

To avoid function rule violations:

• Ensure that each input from the domain maps to a unique output in the codomain.
• Check your ordered pairs carefully to verify that no element repeats as an input with a different output.

Following these guidelines helps maintain clarity and uphold the mathematical definition of a function.