Recitation 8
Math 299
Oct 24, 2013
1. proposition: The sum of the rst n odd numbers is n2 .
?
?
a. Try some examples: 1 + 3 = 22 , 1 + 3 + 5 = 32 , etc.
b. State the Proposition formally in symbols, making clear what is the nth case P (n). How to write a
Recitation 7
Solutions
Math 299
Oct. 17, 2013
1
problem 1. Prove that if x is a positive real number, then x + x 2. (hint: Experiment
starting from the inequality you want to prove and derive a true statement; then try to work
backwards, reversing your ar
Recitation 4
Math 299
Sept. 19, 2013
problem 1. Use truth tables to prove that the statements not (A and B) and (not A)
or (not B) are equivalent.
A
T
T
F
F
B
T
F
T
F
not(A)
F
F
T
T
not(B)
F
T
F
T
A and B
T
F
F
F
not(A and B)
F
T
T
T
not(A) or not(B)
F
T
Math 299
Recitation 5
Sept. 26, 2013
problem 1. Say whether True or False for each of the following and if the following
statement is not true, x the statement to be true.
1.
: Earning a nal grade of C from MTH 299 class is a necessary and sucient
conditi
Math 299
Recitation 9
Oct 31, 2013
1. proposition: For every positive integer n, the polynomial x y divides xn y n .
a. Assume this proposition is true, use it to prove the following: 7 divides 12n 5n , 4 divides 5 7n 3n ,
and 4 divides 3 7n + 5 3n .
Firs
Recitation 11
Math 299
Nov 21, 2013
1. Find the number N such that n > N we have an inequality
3n 1
3 <
n+1
for given as follows:
4
(Sol) > 0, N =
such that n N ,
4
4
3n 1
3n 1 3n 3
4
4
=
=
3 =
<
=
n+1
n+1
n+1
n+1
n
N
(1) = 0.1. : N = 40, that is, for all
Recitation 10
Math 299
Nov 14, 2013
1. Prove that the following system of equations has no integer solutions.
11x 5y = 7
9x + 10y = 3
Hint: Consider each of the equations mod 5.
Solution: Assume, by way of contradiction that (x, y ) Z2 is a solution to th