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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.
 Fall '08
 STAFF
 Math, Calculus

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