12
2.2.
2.2.1 (a) A \ B is the set of students who live within one mile of school and walk to classes.
(b) A [ B is the set of students who live within one mile of school or walk to classes.
(c) A B i
29
4.6.
4.6.2 (a) WXST TSPPYXMSR
(b) NOJK KJGGPODJI
(c) QHAR RABBYHCAJ
4.6.4 We just need to subtract 3 from each letter. For example, E goes down to B, and
B goes down to Y.
(a) BLUE JEANS
(b) TEST T
33
5.1.15 Let P (n) be the statement
n(n + 1)(n + 2)
.
3
The basis step is n = 1, and indeed 1 2 = 123 . For the inductive step, x k
1,
3
and assume that P (k) is true. Then
k(k + 1)(k + 2)
[1 2 + 2 3
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38
6.2.
6.2.1 There are 5 weekdays. These are the pigeonholes. There are six classes. These are
the pigeons. At least d6/5e = 2 classes must be on the same day.
6.2.2 This follows from the pigeonhole
37
6.1.
6.1.1 There are 18 mathematics majors and 325 computer science majors.
(a) By the product rule, there are 18 325 = 5850 ways to pick two representatives
so that one is a mathematics major and
39
6.3.
6.3.1 There are 6 permutations: abc, acb, bac, bca, cab, cba
6.3.2 P(7,7) = 7! = 5040
6.3.3 We want to count permutations of cfw_a, b, c, d, e, f, g that end with a. There are six
letters we n
44
9.5.
9.5.1 (a) This is an equivalence relation. (It is trivial to check reexive, symmetric, and
transitive.)
(b) This is not reexive since it does not contain (1, 1). It is symmetric. It is not
tra
34
5.2.
5.2.3 Let a denote a 3-cent stamp, and let b denote a 5-cent stamp.
(a) 8 = a + b, 9 = 3a, and 10 = 2b, so P (8), P (9), and P (10) are true.
(b) Fix k 10. Assume P (j) is true for all 8 j k.
RULE 1
The Constant Rule
Differentiate the function and multiply by the
constant
RULE 2
The Sum Rule
Differentiate each function separately and
add.
RULE 3
The Difference Rule
Differentiate each fu
20
4.2.
4.2.2 To convert from decimal to binary, we successively divide by 2. We write down the
remainders so obtained from right to left; that is the binary representation of the
given number.
(a) Si
22
4.3.
4.3.2 The numbers 19, 101, 107, and 113 are prime, as we can verify by trial division. The
numbers 27 = 33 and 93 = 3 31 are not prime.
4.3.14 We must nd, by inspection with mental arithmetic,
11
2.1.
2.1.2 There are many correct answers.
(a) cfw_3n | n = 0, 1, . . . , 4
(b) cfw_n 2 Z | 3 n 3
(c) cfw_x 2 English alphabet | x is after l and before q alphabetically
2.1.4 Recall that one set i
13
2.3.
2.3.2 (a) This is not a function because the rule is not well-dened. We do not know
whether f (3) = 3 or f (3) = 3. For a function, it cannot be both at the same
time.
p
(b) This is a function
18
4.1.
4.1.5 We want to show that a | b and b | a, then a = b or a = b. Since a | b, there exists
and integer s such that b = as. Similarly, since b | a, there exists an integer t such
that a = bt. I
9
1.7.
1.7.1 We must show that the whenever we have two odd integers, the sum is even. Suppose
a and b are odd integers. Then there exists integers s and t such that a = 2s + 1 and
b = 2t + 1. Then th
P3: FUNCTIONS
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edu from your UNCG email account with subject line MAT 253 programming assignment
#. Name your le 253s
P2: SETS
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#. Name your le 253s14_yo
P4: MORE FUNCTIONS AND LISTS
Directions. Each project should be submitted via email attachment to [email protected]
edu from your UNCG email account with subject line MAT 253 programming assignment
#. Na
4
1.3.
1.3.2. There are two cases. If p is true, then (p) is the negation of a false proposition,
hence true. Similarly, if p is false, then (p) is also false. Therefore the two
propositions are logic
P1: FOR AND IF
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#. Name your le 253
MATH 310: 1.1 SELECTED SOLUTIONS/HINTS
The question asks us to nd h such that the associated linear system is consistent. Recall
that a linear system is consistent if and only the rightmost column of