1.1. We have at least fear times as many chairs as tables. The number ef
chairs (c) is at least (13} feur times the number ef tables (2'). Hence I? 13 4:.
1.1". Cerrectien ef If 1 and y are neneere real numbers and .1- a y, then
{13%} :1 {IIFJ. If y is n
Math 347 - Spring 2012 - Practice Exam 1 (solutions).
1. Most of the
proofs that we gave in class of 2 is irrational can be modied
to work for 5. Perhaps the easiest is to use the fundamental theorem of
arithmetic.
Assume (to the contrary) that 5 is rati
Math 347. Practice Exam 3.
1. Use the denition of the limit to show that if an L then an is a Cauchy sequence.
2. Prove or nd a counterexample to the following statements.
(a) If an is unbounded, then an has no limit.
(b) If an is not monotone, then an ha
Math 347 - Spring 2012 - Practice Exam 1.
1. Prove that
5 is irrational.
2. Without using words of negation (e.g. not, no, etc.), write the negation of the following statements
(a) For all x R, there exists a y R such that f (x) > y .
(b) For all x N, if
Math 347 - Spring 2012 - Practice Exam 2
1. Give an example of a function with the following properties or explain why no
such function exists.
(a) A function from R to R that is increasing, but is not surjective.
(b) A bijection from Z to the odd integer
Math 347 - Spring 2012 - Practice Exam 2 - Hints/Partial solutions
1. (a) f (x) = 2x
(b) f (x) = 2x + 1
(c) Impossible. A function that has an inverse must be a bijection. Thus it must
be injective.
2. Consider the contrapositive and show that if f (g(x)
Math 347. Exam 1.
Name:
1. Write the negation of the following statements.
(a) There exists an n N, such that for all x R, x2 = n.
(b) For all n N, if n is divisible by 5, then n is divisible by 25.
2. (a) Write the contrapositive of the following stateme
Math 347. Practice Exam 3.
1. Use the denition of the limit to show that if an L then an is a Cauchy sequence.
See Proposition 14.13 in MT
2. Prove or nd a counterexample to the following statements.
(a) If an is unbounded, then an has no limit.
TRUE, c
Math 347. Exam 3.
Name:
1. Prove or nd a counterexample to the following statements.
(a) If xn has a limit and yn does NOT have limit, then xn yn has no limit.
(b) If xn is Cauchy, then there exists s such that |as as+1 | < 1/100.
2. Use the limit denitio
2.2. Anaiysis of If a and i: are integers. then there are integers m. n such
that a = m + n and t3 = m s. The statement is false, since summing the
two equations implies that a necessary condition for the existence of such
integers m. n is thata + i;- b
5.7. There are 12 - 4? + 1 - 48 ways to pick two cards from a standard 52-
card deck such that the rst cord is a spade and the second card is not on
Ace. There are 13 ways to start with a spade. If the spade is not the Ace,
then there are 4'? ways to pick
5.5. Given n married coapies, there are air: 1] ways to form. pairs con-
sisting of one man and one woman who are not married to each other. We
must choose one person of each type. Whichever type we choose rst, we
can choose such a person in :1 ways. Whi
13.5. If the sequence {I} does not converge to zero, then there exists e := I] so
that lt'or uii n, |_r,,| u eFALSE. However, it is true that when {I} does not
converge to zero, there exists e r: D so that for innitely many n, |.1',,| :> c.
13.5. A count
14.1. An unbounded sequence that has no conoergent suhsequence. Let
r" = n. The sequence {x} is unbounded, as are all its subsequences.
An unbounded sequence that has a convergent suhsequence. Let 3:2,, =
and 3.12,, = n for all n. The sequence {y} is un
Math 347. Exam 2.
Name:
1. Give an example of a function with the following properties or explain why no such
function exists (in one or two sentences).
(a) A function from R to R that is injective, but not surjective.
(b) A bijection from Z to R.
2. Let