This is the subset of consisting of all pairs (r, s) of real numbers for which r < s is true. In setbuilder notation, it is the subset . When we write “r < s” we are using the symbol “ < “ as infix notation as described here. Note that saying “it is the subset ” I am describing the set, assuming you are already familiar with “less than”; I am not defining it. If I claimed that was a definition, it would be circular (defining it in terms of itself).
The “ ” relation on is the union of the “equals” and the “less than” relation on .
If A and B are any sets, then the empty set is a subset of and so it fits the definition of a relation from A to B. If this relation is called E, then for every element and , the statement “a E b” is false.
If A and B are any sets, then is a subset of and so it fits the definition of a relation from A to B. This is called the total relation from A to B. If this relation is called T, then for every element and , the statement “a T b” is true.
Many of the examples above use overloaded notation, including the equals, inclusion, empty and total relations.
The set is a subset of , and so it is a relation from the set to the set . If we call this relation , then are all true statements, and all other statements of the form “ ” are false. More about this relation here.
Consider a university at a fixed moment in time. Let S be the set of currently enrolled students and C the set of classes currently offered. The relation E (for “enrolled in”) from S to C is the subset . Then “s E c” means that student s is enrolled in class c. (This can be turned into a mathematical relation by using student numbers as codes for the students and class codes for the classes.)
Let be any positive real number. Then we define a real number r to be near a real number s if and only if | r s | < . So if , then and 3.14 are near each other, but 3.1 and 3.2 are not near each other. Note that nearness depends on a parameter . For a particular parameter , nearness is reflexive and symmetric, but not transitive. (Transitivity is discussed further here.)
Let S be the “is the sister of” relation on the set of all people, and S’ be the “is the sister of” relation on the set of all women. These are not the same relation. For example, S’ is symmetric but S is not.