1873_solutions

and 36 a b ab a b a b a

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Unformatted text preview: Therefore every member of A B must be a subset of A Þ B. c. Is it true that if A and B are transitive then the set A This statement is false. Look at the set , B is transitive?  . d. Is it true that if A is transitive then the successor A  of A (as defined in the preceding exercise) is transitive? This statement is true. Suppose that A is transitive. Every member of A, being a subset of A, must be a subset of A Þ A . Furthermore, the set A is also a subset of A Þ A . Exercises on Set Operations 1. Express the set 2, 3 Draw a figure. 0, 1 as the union of two intervals. 0 2 1 3 We see that 2, 3 0, 1  2, 0 Þ 1, 3 . 2. Given two sets A and B prove that the condition A B is equivalent to the condition A Þ B  B. The right way to approach this sort of problem will depend upon the background and strength of the students. Ideally, one should be able to say that, in any case, B A Þ B and that the condition A Þ B B holds when A B. The alternative is to write two careful proofs, one starting with the assumption A B and the other starting with the assumption A Þ B  B. 3. Given two sets A and B prove that the condition A B is equivalent to the condition A B  A. 4. Given two sets A and B prove that the condition A B is equivalent to the condition A B . 5. Illustrate the identity 37 A BÞC  A by drawing a figure. Then write out a detailed proof. B A C Solution: We obtain the identity A BÞC  A B A C A BÞC A B A C A BÞC A B A C. by showing first that and then showing that To obtain the identity A BÞC AB AC we suppose that x A B Þ C . We know that x A and that x does not belong to the set B Þ C. Thus x A and x cannot belong to either of the sets B and C. In other words, x A B and x A C which tells us that x AB A C . The proof of the assertion A BÞC AB AC is similar and will be left to the reader. A Shorter Solution: A given object x will belong to the set A belongs to neither of the sets B and C. The latter condition says that x and x C which says that x AB A C. B Þ C when x A and x A and x B and also that x A 6. Illustrate the identity A B CA by drawing a figure. Then write out a detailed proof. Use the same figure BÞA C that was used in Exercise 5. As before, we can write out the complete solution or the quick version. The quick version follows: Solution: A given object x will belong to the set A the set B C which means that x B C when x A and x fails to belong to A and x fails to belong to at least one of the sets B and C. The 38 latter condition says that either x x A B Þ A C. A and x B or x A and x C which says that 7. Given that A, B and C are subsets of a set X, prove that the condition A B C  holds if and only if X A Þ X B Þ X C  X. We know that X AÞX BÞX C X. Now if we assume that A B C  then every member of X must fail to belong to at least one of the sets A and B and C, which tells us that X X A Þ X B Þ X C. Thus if we are given that A B C  then the equation X A Þ X B Þ X C X must hold. We now assume that the equat...
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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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