A relation is a set of ordered pairs that connects elements from one set (the domain) to elements in another set (the range). Relations are a fundamental concept in mathematics, particularly in the study of functions and their properties.
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A relation can be represented in various ways, including as a set of ordered pairs, a table, a graph, or a mapping diagram.
Vertical line test: A relation is a function if and only if no vertical line intersects the graph of the relation more than once.
Horizontal line test: A function is one-to-one if and only if no horizontal line intersects the graph of the function more than once.
Inverse relations: If a relation $R$ is a function, then the inverse relation $R^{-1}$ is also a function, with the domain and range of $R^{-1}$ swapped.
Composition of relations: If $R$ is a relation from set $A$ to set $B$, and $S$ is a relation from set $B$ to set $C$, then the composition $S \circ R$ is a relation from set $A$ to set $C$.
Review Questions
Explain the relationship between a relation and a function, and describe how the vertical line test can be used to determine if a relation is a function.
A function is a special type of relation where each element in the domain is paired with exactly one element in the range. The vertical line test can be used to determine if a relation is a function: if no vertical line intersects the graph of the relation more than once, then the relation is a function. This is because a function must have a unique output value for each input value, and the vertical line test ensures that there is only one point of intersection between the graph and any vertical line.
Describe the concept of an inverse relation and explain how it is related to the horizontal line test for one-to-one functions.
If a relation $R$ is a function, then the inverse relation $R^{-1}$ is also a function, with the domain and range of $R^{-1}$ swapped. The horizontal line test can be used to determine if a function is one-to-one: if no horizontal line intersects the graph of the function more than once, then the function is one-to-one. This is because a one-to-one function must have a unique input value for each output value, and the horizontal line test ensures that there is only one point of intersection between the graph and any horizontal line. The horizontal line test for one-to-one functions is related to the inverse relation, as a function is one-to-one if and only if its inverse relation is also a function.
Explain the concept of composition of relations and how it can be used to combine multiple relations to create a new relation.
The composition of relations is a way to combine multiple relations to create a new relation. If $R$ is a relation from set $A$ to set $B$, and $S$ is a relation from set $B$ to set $C$, then the composition $S \circ R$ is a relation from set $A$ to set $C$. The composition operation allows you to take the output of one relation and use it as the input for another relation, effectively chaining the relations together. Composing relations is a powerful tool in mathematics, as it enables you to build complex relationships from simpler ones, and is particularly important in the study of functions and their properties.