[answered] MATH 109 FALL 2016 SAMPLE MIDTERM 2 Instructions: Justify a

first and final questions

1 (10 pts). Let R? = R {0} be the set of all nonzero real numbers. Consider the functionf : R? ? R? defined by f(x) = 2/x.Prove that f is bijective. Find a formula for its inverse function f?1 and justify that yourformula is correct

A set of 100 voters are asked about how they plan to vote on three statewide ballotpropositions, Propositions A, B, and C. On each proposition, they can vote either yes or no.In the survey, 40 said they plan to vote yes on A, 50 said they will vote yes on B, and 60said they will vote yes on C. 30 voters plan to vote for both A and B, 10 plan to vote forboth A and C, 5 voters plan to vote yes on all three propositions, and 5 voters plan to voteno on all three propositions.How many voters are there that are planning to vote yes on B but no on both A and C?

MATH 109 FALL 2016 SAMPLE MIDTERM 2

 

Instructions: Justify all of your answers, and show your work. You may use the result

 

of one part of a problem in the proof of a later part, even if you do not complete the proof

 

of the earlier part. You may quote basic theorems proved in the textbook or in class, unless

 

the problem says otherwise, or unless reproving the result of the theorem is the point of the

 

problem. Do not quote the results of homework exercises.

 

1 (10 pts). Let R? = R \ {0} be the set of all nonzero real numbers. Consider the function

 

f : R? ? R? defined by f (x) = 2/x.

 

Prove that f is bijective. Find a formula for its inverse function f ?1 and justify that your

 

formula is correct.

 

2 (10 pts). Let f : A ? B and g : B ? C be functions, and consider the composition

 

h = g ? f : A ? C.

 

(a). Show that if f and g are both injective, then h is injective.

 

(a). Show that if h is injective, then f is injective. Give an example showing that g does

 

not have to be injective.

 

3 (10 pts). Consider the decimal expansions of real numbers x with 0 < x < 1. For each

 

such x, assume the result that x has a unique decimal expansion of the form x = .a0 a1 a2 a3 . . .

 

a1

 

an

 

with ai ? {0, 1, . . . , 9}, (that is, x = a100 + 100

 

+ ? ? ? + 10

 

n + . . . ) and where it is not the case

 

that there is some N such that ai = 9 for all i ? N .

 

Show that the set X = {x ? R|0 < x < 1} is uncountable. (This is a theorem in the book.

 

You are expected to reprove it here.)

 

4 (10 pts).

 

A set of 100 voters are asked about how they plan to vote on three statewide ballot

 

propositions, Propositions A, B, and C. On each proposition, they can vote either yes or no.

 

In the survey, 40 said they plan to vote yes on A, 50 said they will vote yes on B, and 60

 

said they will vote yes on C. 30 voters plan to vote for both A and B, 10 plan to vote for

 

both A and C, 5 voters plan to vote yes on all three propositions, and 5 voters plan to vote

 

no on all three propositions.

 

How many voters are there that are planning to vote yes on B but no on both A and C? Date: November 3, 2016.

 

1

 

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