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Unformatted text preview: f Set Theory
Some Elementary Exercises on Sets
1. Given objects a, b and y and given that a, b a, y , prove that b y. Solution: From the fact that a, b a, y and the fact that b
a, b we deduce that b
a, y .
Therefore there are two possibilities; either b a or b y. In the event that b a then the equation
a, b a, y becomes a a, y and we conclude that y
a which tells us that y a. So in this
case we have a b y. So in either case we know that b y. 2. Prove that if a, b, x and y are any given objects and if a, b x, y then either a x and b y, or a y
and b x.
Since a
a, b and a, b x, y we know that a
x, y . Therefore either a x or a y. In the
event that a x we have a, b a, y and it follows from Exercise 1 that b y. Similarly, if a y
then b x.
3. Prove that if a, b, x and y are any given objects and if
a , a, b x , x, y ,
then a x and b y.
The preceding exercises guarantee that if
a , a, b x , x, y ,
then either a x or a, b x, y from which we can see at once that a x and b y.
4. Describe the set p .
Since the only subset of the set is itself we have p 5. Describe the set p p
.
You should be able to prove that p p . . , 6. Given that A a, b, c, d , list all of the subsets of A. Describe the set p A .
You should be able to show that p a, b, c, d is
, a , b , c , d , a, b , a, c , a, d , b, c , b, d , c, d , a, b, c , a, b, d , a, c, d , b, c, d , a, b, c, d
7. Given that A a, b , list all of the subsets of A. Describe the set p p A .
Since p A
, a , b , a, b
the set p p A is
,
b , a, b a ,
, b , a, b , ,a,b , , ,a , a , a, b ,
, ,b , b , a, b 8. Given any set S, the successor of S is the set S defined by
S S Þ S .
a. Describe the sets , , and . 36 , , a, b
, , a,b a , b , a, b ,
, a , a, b , , a , b , a, b
Þ Þ
Þ
Þ
Þ
Þ ,
, b. Is it true that if A B then A
B?
This assertion is false. Note that
,
. ,
, , , , but the set ,
c. Given that A and B are sets and that at least one of the conditions A
that A B , prove that A B. ,
, , , ,
is not a subset of the set B and B A is false, and given Solution: We are given that A Þ A B Þ B . We shall assume that the condition A B is
false. The other case B A is analogous to this one. Now since A A Þ A B Þ B we know
that either A B or A
B . We have assumed that A does not belong to B and so the condition
A
B must hold; and we conclude that A B. 9. A set A is said to be transitive if every member of A is a subset of A.
a. Is it true that if A and B are transitive then the set A Þ B is transitive?
This statement is true. Any member of A has to be a subset of A and is certainly a subset of
A Þ B. The same argument can be applied to any member of B.
b. Is it true that if A and B are transitive then the set A B is transitive?
This statement is true. A member of A B, being a member of A, must be a subset of A and,
being a member of B, must be a subset of B....
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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