Math 350: Foundations
Fall 2013
HW 1: Selected Solutions
1.1/3, 4; 1.2/35; 1.3/25, 7
1.1.4 Ayumi and Coco are mathematicians, and Bethesda is a dalek. We begin
the proof by rst showing that Bethesda is a dalek. Suppose otherwise;
then Bethesda is a mathem
Math 350: Foundations
Fall 2013
HW 4: Selected Solutions
Prof. Shahed Sharif
2.6/4, 6; 3.1/3, 5, 7, 8; 3.2/5, 6; 3.3/2, 4, 7, 9, 11, 14
3.1.7
(a) To construct an 8-digit binary string, we make 8 independent choices
(one for each digit), each of which has
Math 350: Foundations
Fall 2013
HW 3: Selected Solutions
Prof. Shahed Sharif
2.1/3-5; 2.2/4-7; 2.3/3e-i, 4, 5, 8-12, 14
2.1.5
Claim. The cardinalities of the given sets are:
(a) 13
(b) The cardinality is innite.
(c) 45
(d) 333
(e) 1
Proof. (a) The number
Math 350: Foundations
Fall 2013
HW 2: Selected Solutions
Prof. Shahed Sharif
1.4/3; 1.5/46, 8, 9; 1.6/2, 3; 1.7/1, 2
1.4.3
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
Q or R
T
T
T
F
T
T
T
F
If P, then Q or R
T
T
T
F
T
T
T
T
1.6.2
Claim. Acidophi
Math 350: Foundations
Fall 2013
HW 6: Selected Solutions
Prof. Shahed Sharif
5/2, 5, 6, 11, 17, 19, 23, 24; 6/2, 4, 8, 9, 12, 14
5.11 Proof. We will prove the contrapositive. Suppose the statement x is
even or y is odd is not true. By DeMorgans law this m
Math 350: Foundations
Fall 2013
HW 7: Selected Solutions
Prof. Shahed Sharif
7/2, 5, 8, 15, 18, 22, 28; 8/2, 5, 7, 23, 30; 9/4, 7, 8
7.8 Proof. First, we prove that a b (mod 10) implies a b (mod 2) and
a b (mod 5). Assume a b (mod 10), so 10 | ( a b) by d
Math 350: Foundations
Fall 2013
HW 11: Selected Solutions
Prof. Shahed Sharif
12.4/5, 7, 9; 12.5/2, 6, 7; 12.6/2, 6, 8, 11; 13.1/1, 3, 4, 11; 13.2/3-5
12.4.9 The formulas are ( g f )(m, n) = (m + n, m + n) for (m, n) Z Z, and
( f g)(m) = 2m for m Z
12.5.6
Math 350: Foundations
Fall 2013
HW 10: Selected Solutions
Prof. Shahed Sharif
12.1/1, 2, 5-11; 12.2/2, 3, 5, 6, 11-14
12.1.2 The domain of f is cfw_ a, b, c, d, the range of f is cfw_2, 3, 4, 5, and f (b) = 3,
f (d) = 5.
12.1.8 The relation f is not a fun
Math 350: Foundations
Fall 2013
HW 9: Selected Solutions
Prof. Shahed Sharif
City blocks problem (CBP); 11.2/2, 3, 8, 11-13; 11.3/4; 11.4/2, 4-6, 8
CBP See City blocks solution
11.2.11 The claim is false. A trivial counterexample is the relation R = A
A.
Math 350: Foundations
Fall 2013
HW 8: Selected Solutions
Prof. Shahed Sharif
9/13, 21, 25, 30; 10/2, 4, 8, 9, 18, 21, 23, 25
9.21 The claim given here is false. To disprove the claim, we assume it is true
and nd a contradiction.
Proof. Let p, q be prime n
Math 350: Foundations
Fall 2013
HW 5: Selected Solutions
Prof. Shahed Sharif
3.4/2, 3, 5, 6, 9; 4/4-6, 10, 15, 24, 26
3.4.5 Set x = y = 1 in the binomial theorem and simplify the resulting equation.
4.5 Proof. Assume x is even. Then x = 2a for some a Z by