**Unformatted text preview: **1832832885 11:13 5188422415 LICE MtﬂtIN LIERﬂtR‘r" F'tﬂtGE 81.-"'81 George M. Bergman. Fall 2005. Math 55 31 October. 2005
100 Lewis Hall Second Midterm 1:10-2:00 1. (24 points. 8 points each.) Short answer questions. A correct answer will get full _
credit whether or not work is shown. An incorrect answer may get partial credit if work is
given that follows a basically correct method. (a) How many S-element subsets does a IOU—element set have? You may express your
answer in any of the forms used in the text. (b) In how many ways can one put together a packet of 30 pieces of fruit. if one has 5
kinds of fruit in unlimited supply. and all that matters is how many of each. kind go into
the packet? Again. you may express yOut‘ answer in any of the forms used in. the text. (c) What is the probability that an integer chosen at random from {0. 1.. 10} (where all
members of this set have'equal probability of being chosen) is odd? (Note: Do not confuse {o.1.....10} with {l.....10}.) 2. (24 points. 8 points each.) Complete the following deﬁnitions. Your deﬁnitions do not
have to have the same wording as those in the text. but for full credit they should be clear.
and be equivalent in meaning to those. (a) Integers m anti n are said to be relatively prime if . . .
(b) A set S is said to be cotmrnbie if . . . (c) The lexicographic order on the set of length-rt strings of integers is deﬁned by
considering a string to] .... . on) to precede a string (b1 . . it”) under this order if . . . 3. (24 points. 3 points each.) For each of the items listed below, either give an example
with the properties stated, or give a brief reason why no such exompie exists. If you give an example. you do not have to prove that it has the property stated;
however. your examples should be speciﬁc; i.e.. even if there are many objects of a given
sort, you should name a particular one. If you give a reason why no example exists. don’t
worry about giving reasons for your reasons; a simple statement will sufﬁce. (at) Two1 {integers o. and b. neither of which is divisible by 17. such that
6 e it 1'(mod 17). (b) A program which never halts. (Use pseudocode if you give an example.) (e) A one—toeone function from the set of ordered pairs (rt. 5:) with a. bell. . 5} to the
set of ordered pairs (c. d) with CE{1,... .6}. de{1,.4}. 4. (28 points. 14 points each.) Short proofs. I am giving you a page for 630-11, in C386
some of you give reundabout proofs or have several false starts. But concise proofs should
take less than half a page each. (a) Show that if a. b. A. B are integers and m a positive integer. with a E A (mod m)
and b g 13' (mod m). then a}: E AB (mod m). Since this is a result proved in Rosen
(though with slightly different notation). you. may not call on. that result. or results proved
from it. in your proof of this statement. (b) Prove that for every nonnegative integer it. one has 2:320 i2”- = (”—1)2""+1 .1. 2_ ...

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