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Math week 2

4.1 5a. Consider the following program segment (written in pseudocode):

for i := 1 to 123 do

for j := 1 to i do

print i * j

a) How many times is the print statement of the third line

executed? 7626

4.2 18. Consider the permutations of 1, 2, 3, 4. The permutation

1432, for instance, is said to have one ascent—namely, 14

(since 1 < 4). This same permutation also has two descents—

namely, 43 (since 4 > 3) and 32 (since 3 > 2). The permutation

1423, on the other hand, has two ascents, at 14 and 23—and

the one descent 42.

a) How many permutations of 1, 2, 3 have k ascents, for

k _ 0, 1, 2?

123:2 ascents

132:1 ascent

213:1 ascent

231:1 ascent

312: 1 ascent

321: o ascent

So the permutation of (1,2,3) have 1 ascent of 2, 4 acents of 1 and 1 zero ascent.

4.3 4. If a, b, c ∈ Z+ and a|bc, does it follow that a|b or a|c?

Yes it does because a/bc implies that bc can be a multiple of a

However it can’t be a factor of ab because it would have to be a factor of b or a.

4.4 1. For each of the following pairs a, b ∈ Z+, determine

gcd(a, b) and express it as a linear combination of a, b.

a) 231, 1820

gcd(1820, 231) = 7 =1820(8) + 231(−63)

5.1 4. For which sets A, B is it true that A _ B _ B _ A?

5.2 4. If there are 2187 functions f : A→B and |B| _ 3, what

is |A|?

[pic] is the number of functions from A to B

2187 = 3^|A|

then you have to solve for |A| and you get

|A| = 7

5.3 1. Give an example of finite sets A and B with |A|, |B| ≥ 4

and a function f : A→B such that

a) f is neither one-to-one nor onto;

answer A _ {1, 2, 3, 4}

5.4 13. Let Ai , 1 ≤ i ≤ 5, be the domains for a table D ⊆ A1 _

A2 _ A3 _ A4 _ A5, where A1 _ {U, V,W, X, Y, Z} (used as

code names for different cereals in a test), andA2 _ A3 _ A4 _

A5 _ Z+. The table D is given as Table 5.7.

a) What is the degree of the table?

The answer is 5

Exercise 5.7, problem 1a 293

1. Use the...