Problem 78 - Introductory Algebra for College Students

Question

Here are two sets of ordered pairs: $$\begin{array}{l} \text { set } 1:\\{(1,5),(2,5)\\} \\ \text { set } 2:\\{(5,1),(5,2)\\} \end{array}$$ In which set is each \(x\) -coordinate paired with one and only one \(y\) -coordinate?

Step-by-Step Solution

Answer

Set 1 is the set where every x-coordinate is paired with one and only one y-coordinate.

Step 1: Examine Set 1

In the first set, \((1,5)\) and \((2,5)\), the x-coordinates are unique for each ordered pair, meaning each x-coordinate uniquely correlates to a y-coordinate. This set demonstrates a one-to-one relationship.

Step 2: Examine Set 2

In the second set, \((5,1)\) and \((5,2)\), the x-coordinate, which is 5, is the same for both ordered pairs. Consequently, this x-coordinate is paired with multiple y-coordinates (1 and 2), meaning it does not have a one-to-one relationship.

Key Concepts

One-to-One Relationship

Understanding a one-to-one relationship in the context of ordered pairs is essential for grasping the fundamentals of functions in algebra. Imagine pairs of shoes and socks; each shoe (x-coordinate) is designed to be worn with just one specific sock (y-coordinate), and vice versa.

An ordered pair, \(x, y\), can be pictured as a unique pairing where each x-coordinate matches with one, and only one, y-coordinate. This exclusivity defines a one-to-one relationship. In Set 1, consisting of \(1,5\) and \(2,5\), each x-value is distinct and correlates to a single y-value, thus fulfilling the criteria of a one-to-one relationship.

In contrast, Set 2, containing \(5,1\) and \(5,2\), violates this rule as the x-coordinate 5 is paired with more than one y-coordinate. Envision two toothbrushes (the x-value of 5) designated to be used by two different people (y-values 1 and 2); this clearly is not a one-to-one 'brush-to-owner' relationship.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot ordered pairs \(x, y\). It's split into four quadrants by a horizontal line (x-axis) and a vertical line (y-axis). Each point on this plane is identifiable by its unique position in relation to these two axes.

When we map our ordered pairs from Set 1 and Set 2 onto the coordinate plane, each pair \(x, y\) finds its own specific location. For instance, the pair \(1,5\) from Set 1 would be located to the right of the y-axis, and above the x-axis in Quadrant I. Its counterpart from Set 2, \(5,1\), would be positioned in the same quadrant, but this time further right along the x-axis, and closer to the x-axis along the y-direction.

Visualization on the coordinate plane can make it easier to distinguish between sets that exhibit a one-to-one relationship and those that do not. By plotting, we can see at a glance whether an x-coordinate is linked to multiple y-coordinates, which is not allowed in a one-to-one relationship.

Algebraic Relationships

Algebraic relationships describe how variables interact with each other. These interactions can often be represented through equations or sets of ordered pairs, which show how changing one variable affects another.

In the case of Set 1 and Set 2, we're exploring a foundational concept of algebraic relationships called mapping. Specifically, we're seeing how one variable (the x-value) corresponds to another (the y-value). A one-to-one algebraic relationship assures that for every input, there is a unique output, reinforcing the concept of function in mathematics.

The algebraic approach allows us to analyze patterns, predict outcomes, and understand the connections between quantities. When the sets fail to demonstrate a one-to-one relationship, such as Set 2, we face a scenario that isn't fitting for functions that are strict in their pairings, reminding us that not all algebraic relationships can be categorized as one-to-one functions.