Problem 127 - College Algebra Essentials

Question

Give an example of a relation with the following characteristics: The relation is a function containing two ordered pairs. Reversing the components in each ordered pair results in a relation that is not a function.

Step-by-Step Solution

Answer

A function that satisfies the conditions given by the exercise is \( f = \{(1,2), (3,4)\} \). When we reverse the order of the pairs, \( R = \{(2,1), (4,3)\} \), which is not a function as no element in the domain maps to more than one element in its codomain.

Step 1: Construct a function

First, we must construct a function that contains two ordered pairs. An example of such a function would be \( f = \{(1,2), (3,4)\} \). The function \( f \) is defined so that it maps 1 to 2 and 3 to 4.

Step 2: Reverse the order of the pairs

Next, we reverse the order of the pairs in our function \( f \) to create a new relation \( R \). This means \( R = \{(2,1), (4,3)\} \).

Step 3: Check if the relation is still a function

Finally, we verify whether the relation \( R \) is still a function or not, according to the definition of a function. Since we only reversed the existing pairs without creating duplicate mappings for elements in the domain, the relation \( R \) is not a function, fulfilling the conditions set by the exercise.