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Lecture 9
Relations and their properties
Let A and B be sets. A binary relation from A to B is a subset of 
. In other words, a binary relation from A to B is a set R of ordered pairs where the first element of each ordered pair comes from A and the second element comes from B. We use the notation aRb to denote that (a, b) Î R and 
to denote that (a, b) Ï R. Moreover, when (a, b) belongs to R, a is said to be related to b by R. Binary relations represent relationships between the elements of two sets. We will omit the word “ binary ” when there is no danger of confusion.
Example. Let A be the set of all cities, and let B be the set of the 50 states in the USA. Define the relation R by specifying that (a, b) belongs to R if city a is in state b. For instance, (Boulder, Colorado), (Chicago, Illinois), (Cupertino, California) are in R, but (Chicago, Colorado), (Cupertino, Illinois) are not in R.
Functions as relations. Recall that a function f from a set A to a set B assigns a unique element of B to each element of A. The graph of f is the set of ordered pairs (a, b) such that b = f(a). Since the graph of f is a subset of , it is a relation from A to B. Moreover, the graph of a function has the property that every element of A is the first element of exactly one ordered pair of the graph. Conversely, if R is a relation from A to B such that every element in A is the first element of exactly one ordered pair of R, then a function can be defined with R as its graph. This can be done by assigning to an element a of A the unique element 
such that (a, b) Î R.
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