Problem 10 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

Yes, the set of ordered pairs \{(a, 1),(b, 2),(c, 3)\} represents a function from A to B.

Step 1: Understand the concept of a function

A function from A to B is a set of ordered pairs where each element from A is associated with exactly one element from B.

Step 2: Examine the given set of ordered pairs

Here we have the set of ordered pairs \{(a, 1),(b, 2),(c, 3)\}. From this, we can see that each element in A (i.e., a, b, c) is associated with exactly one element in B (i.e., 1, 2, 3). This suggests that the given set of pairs could be a function from A to B.

Step 3: Confirm the presence of an ordered pair for each element of A

On further analysis, there is indeed one and only one ordered pair for each element in set A. This confirms that the set of ordered pairs indeed represents a function from A to B.

Key Concepts

Ordered Pairs in Mathematics

Understanding ordered pairs is foundational in comprehending how relations and functions work in mathematics. An ordered pair consists of two elements where the order is significant. Typically, they are represented as \((x, y)\), where \(x\) is the first component and \(y\) is the second. This order matters because \((x, y)\) is different from \((y, x)\). Such pairs help to establish connections between two sets of values.

For example, in geometry, points on a coordinate plane are expressed as ordered pairs, linking the x-coordinate with the y-coordinate. In the exercise above, the ordered pairs \((a, 1), (b, 2), (c, 3)\) show a relationship between elements of set \(A\) and \(B\).

Ordered pairs are fundamental in defining functions since they clearly show which element from one set is paired with an element from another. Recognizing that each element from set \(A\) is linked to a unique element in set \(B\) confirms the function's requirements.

Set Theory Fundamentals

Set theory is a branch of mathematical logic that deals with collections of objects, known as sets. This theory provides the groundwork for various mathematical concepts, including ordered pairs and relations.

A set is any collection of distinct objects, considered as an object in its own right. Sets can be finite, like \(A=\{a, b, c\}\) in the given exercise, or infinite, depending on their elements. Elements within a set are unordered, which differentiates sets from ordered pairs.

To understand the relation from \(A\) to \(B\), it's essential to recognize how sets are used. Sets allow us to scientifically track and analyze elements that satisfy certain conditions or rules, like pairing elements from two different sets in a function. This is evident in functions where each element in the domain (set \(A\) here) maps to a single element in the codomain (set \(B\) here). Understanding these relations requires a grasp of basic set theory principles.

Relation from Set A to Set B

In mathematics, a relation from one set to another describes how elements from the first set are connected or related to elements in the second set. Given sets \(A=\{a, b, c\}\) and \(B=\{0, 1, 2, 3\}\), a relation from \(A\) to \(B\) involves pairing each element of \(A\) with one or more elements of \(B\).

However, for this pairing to be a function, each element in \(A\) must be associated with exactly one element in \(B\). In the exercise, the set \(\{(a, 1), (b, 2), (c, 3)\}\) is examined to determine if it's a function. Every element in \(A\) appears only once and is paired with a single element in \(B\), confirming it's indeed a function from \(A\) to \(B\).

Examining relations helps students grasp how different sets interact, which is key in numerous mathematical contexts. Understanding how to evaluate whether these relations represent functions requires assessing the ordered pairs to ensure they meet this one-to-one pairing condition.