Explain the words relation and function. Include in your explanation how the meanings of the two words differ. Give a real-world example of a relation that is not a function. Is it possible to give an example of a function that is not a relation? If so, give one, if not, explain why it is not possible.
y^2 = x
y is not a function of x, since there are two values of y for eacvh value of x>0
Any function is also a relation.
Sure! Let's start by explaining the words "relation" and "function" and how they differ.
In mathematics, a relation is a set of ordered pairs that relate elements from two or more sets. It represents a connection or association between the elements. For example, suppose we have two sets: set A {1, 2, 3} and set B {4, 5}. We can define a relation R as a set of ordered pairs where the first element belongs to set A, and the second element belongs to set B. So, R = {(1, 4), (2, 5)}. This relation establishes a connection between elements of A and elements of B.
On the other hand, a function is a specific type of relation where each element from the domain (input) set is assigned to exactly one element in the codomain (output) set. In simpler terms, every input value has a unique output value. For instance, consider a function F that maps the set of integers to their squares. So, F = {(1, 1), (2, 4), (3, 9)}. Here, each input (1, 2, 3) is associated with only one output (1, 4, 9), making it a function.
Now, let's address the example of a relation that is not a function. One such example is a relation that associates a person's name with their favorite colors. Suppose we have a set of names {John, Sarah, Mike} and a set of colors {Blue, Red, Blue}. The relation R = {(John, Blue), (Sarah, Red), (Mike, Blue)} connects the names with their favorite colors. In this case, since both John and Mike have the same favorite color (Blue), the relation is not a function since one input (favorite color) is associated with multiple outputs (names).
As for your second question, it is not possible to provide an example of a function that is not a relation. This is because every function implies a relation between the domain and codomain. In other words, a function is a specific type of relation that follows the rule of having exactly one output for each input. So, every function can also be considered a relation, but not every relation can be considered a function.
I hope this explanation clarifies the meanings of relation and function and demonstrates their differences. Let me know if you have any further questions!