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Unformatted text preview: 38 Chapter 2 Basic Structures: Sets, Functions, Sequences, Sums, and Matrices CHAPTER 2 Basic Structures: Sets, Functions, Sequences, Sums, and Matrices SECTION 2.1 Sets 2. There are of course an infinite number of correct answers. a) { 3 n  n = 0 , 1 , 2 , 3 , 4 } or { x  x is a multiple of 3 ∧ ≤ x ≤ 12 } . b) { x   3 ≤ x ≤ 3 } , where we are assuming that the domain (universe of discourse) is the set of integers. c) { x  x is a letter of the word monopoly other than l or y } . 4. Recall that one set is a subset of another set if every element of the first set is also an element of the second. a) The second condition imposes an extra requirement, so clearly the second set is a subset of the first, but not vice versa. b) Again the second condition imposes an extra requirement, so the second set is a subset of the first, but not vice versa. c) There could well be students studying discrete mathematics but not data structures (for example, pure math majors) and students studying data structure but not discrete mathematics (at least not this semester— one could argue that the knowing the latter is necessary to really understand the former!), so neither set is a subset of the other. 6. Each of the sets is a subset of itself. Aside from that, the only relations are B ⊆ A , C ⊆ A , and C ⊆ D . 8. a) Since the set contains only integers and { 2 } is a set, not an integer, { 2 } is not an element. b) Since the set contains only integers and { 2 } is a set, not an integer, { 2 } is not an element. c) The set has two elements. One of them is patently { 2 } . d) The set has two elements. One of them is patently { 2 } . e) The set has two elements. One of them is patently { 2 } . f) The set has only one element, {{ 2 }} ; since this is not the same as { 2 } (the former is a set containing a set, whereas the latter is a set containing a number), { 2 } is not an element of {{{ 2 }}} . 10. a) true b) true c) false—see part (a) d) true e) true—the one element in the set on the left is an element of the set on the right, and the sets are not equal f) true—similar to part (e) g) false—the two sets are equal 12. The numbers 1, 3, 5, 7, and 9 form a subset of the set of all ten positive integers under discussion, as shown here. Section 2.1 Sets 39 14. We put the subsets inside the supersets. Thus the answer is as shown. 16. We allow B and C to overlap, because we are told nothing about their relationship. The set A must be a subset of each of them, and that forces it to be positioned as shown. We cannot actually show the properness of the subset relationships in the diagram, because we don’t know where the elements in B and C that are not in A are located—there might be only one (which is in both B and C ), or they might be located in portions of B and/or C outside the other. Thus the answer is as shown, but with the added condition that there must be at least one element of B not in A and one element of C not in A ....
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This note was uploaded on 03/21/2012 for the course CMPSC 40 taught by Professor Egiceoclu during the Spring '09 term at UCSB.
 Spring '09
 Egiceoclu

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