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Both the domain and range of r are the entire set S. yx . 3. Suppose that S is the set of all integers n such that 2 n 20 and that r is the relation in S that consists of all
pairs x, y for which x is a factor of y but x y. What are the domain and range of r?
4. Suppose that r is the relation in R that contains all the ordered pairs x, y satisfying x 2 4y 2 1. What are
the domain and range of r? Solution: A real number x is in the domain of the relation r defined in this exercise if and only if it is
possible to find a real number y such that x 2 4y 2 1. Such a real number y will exist if and only if
1 x 2 0 and to the domain of the relation r is the interval 1, 1 . In a similar way we can see that the
range of the relation r is the interval 4, 4 . 5. Suppose that r is the relation in R that contains all the ordered pairs x, y satisfying x 2 4y 2
the domain and range of r?
6. Suppose that r is the relation in R that contains all the ordered pairs x, y satisfying x 2
the domain and range of r? 1. What are 4y 2 1. What are 7. Is it true that if r 1 and r 2 are relations in a set S then the domain of the relation r 1 Þ r 2 is the union of the
domains of r 1 and r 2 ?
The assertion is obviously true.
8. Suppose that
Show that the relations and S ,
,,
are the same in S? , , , , . 9. Suppose that is a collection of families of sets and that is nested. Suppose that for every member of the
collection the relations and are the same in . Prove that the relations and are the same in the
family Þ .
Suppose that A and B belong to Þ and choose a member of to which both A and B belong.
Since the relations and are the same in , the condition A B is equivalent to the condition 43 A B. Exercises on Equivalence Relations
1. For each of the relations given in the list of examples, determine whether or not the relation is reflexive,
whether or not it is symmetric and whether or not it is transitive.
2. Give an example of a relation that is reflexive and symmetric but not transitive.
We define
r x, y
R R x y  1 .
3. Give an example of a relation that is reflexive and transitive but not symmetric.
The relation in the power set p R of R is reflexive and transitive but not symmetric.
4. Give an example of a relation that is symmetric and transitive but not reflexive.
Take the relation
R R x 1 and y 1 .
r x, y
5. Suppose that r is a relation in a set S and that r satisfies the following three conditions:
a. For every x S there exists a member y S such that x r y. b. The relation r is symmetric
c. The relation r is transitive.
Prove that r is an equivalence relation.
All we have to do is prove that r is reflexive. Suppose that x S. Using condition a, choose a
member y of S such that x r y. Since r is symmetric we know that y r x and therefore, since r is
transitive we have x r x.
6. Suppose that ß is an equivalence relation in a set S and that E is a subset of S that contains precisely one
member of each equivalence class of the relation ß. Prove that...
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 Fall '08
 STAFF
 Math, Calculus

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