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 |AÈB| = |A| + |B|.

2. If A and B are any multisets, then |AÈB| = |A| + |B|.

3. If A, B, and C are any multisets, then A È (B Ç C) = (A È B) Ç (A È C).

4. If A and B are any sets, then |AÇB| + |A È B| = |A| + |B|.

5. If A and B are any multisets, then |AÇB| + |A È 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 ?)

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