1873_solutions

A we have xgfx b given that there exists a function h

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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.

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