Download Math 58 - Number Theory

Math 58 - Number Theory
September 26, 2006
Homework 7 (due September 28):
Mark each of the statements 1-10 is TRUE or FALSE, and if you have any doubts, explain.
[Note: If A is a set, then |A|, called its cardinality, is the number of elements in A. If A is a
multiset, then |A| --- still called the cardinality --- is the sum of the multiplicities of all of
the elements of A; for example, |{{1, 2, 3, 3 }}| = 4. ]
1. If A and B are any sets, then |AB| = |A| + |B|.
2. If A and B are any multisets, then |AB| = |A| + |B|.
3. If A, B, and C are any multisets, then A  (B  C) = (A  B)  (A  C).
4. If A and B are any sets, then |AB| + |A  B| = |A| + |B|.
5. If A and B are any multisets, then |AB| + |A  B| = |A| + |B|.
6. If A, B, and C are any sets of numbers, then (A + B) + C = A + (B + C).
7. If A and B are any sets of numbers, then 2(A+B) = (2A) + (2B).
8. If A, B, and C are any sets of numbers for which A + C = B + C, then A = B.
9. If A is any set of numbers, then A + 2A = 3A.
10. If S is the set of all integer multiples of 5, then S + 2S = 3S.
(end of true-false questions)
11. Describe the set (11Z + 5)(11Z + 6).
12. Let S be the set of squares of integers; that is, S = {0, 1, 4, 9, 16, … }. List the smallest 20
elements of each of these sets.
S+S
S + 2S
S+S+S
(Extra challenge: Which of these sets, if any, contain the integer 1007 ?)
(end)