- Draw a Venn Diagram showing the relationship between the following sets: (4 points)
A = The set of rational numbers
B = The set of integers
C = The set of real numbers
D = The set of natural numbers
- Of the four sets just mentioned, determine which contain the following numbers: (8 points)
- Write each sequence, filling in the missing numbers. Then describe the number pattern. (16 points)
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1, 3, 6, 10, 15, 21, __, __ |
29, 23, 17, 11, 5, __, __ |
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1, 4, 9, 16, 25, 36, __, __ |
1, 1, 2, 3, 5, 8, 13, __, __ |
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1, 4, 16, 64, 256, __, __ |
16/3, 8, 12, 18, 27, __, __ |
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2, 3, 5, 9, 17, 33, ___, ___ |
1, 8, 27, 64, 125, ___, ___ |
- A consumer survey was conducted to examine patterns in ownership of microwave ovens, answering machines, and VCRs. Use a Venn Diagram to figure out how many people were surveyed. (8 points)
200 owned microwaves
290 owned answering machines
340 owned VCRs
130 owned microwaves and answering machines
200 owned answering machines and VCRs
170 owned microwaves and VCRs
110 owned all three
60 owned none
- Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A, B, and C be subsets of S such that A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5, 6}, and C = {5, 6, 8, 9}.
- What is the union of A and B? (3 points)
- What is the intersection of B and C? (3 points)
- What is the complement of C? (3 points)
- Draw a Venn Diagram showing the locations of elements 1 to 10. (5 points)
The set of prime numbers consists of all natural numbers that have exactly two unique factors (numbers that divide into it evenly).
- For example:
- 1 is not prime since its only factor is itself
2 is prime since it has exactly two factors (1 and 2)
3 is prime since it has exactly two factors (1 and 3)
4 is not prime since it has three factors (1, 2, and 4)
- List the first twenty prime numbers. (5 points)
- There is a mathematical proof that shows that the number of primes is infinite. Use the hints given in class to explain the proof. (20 points)
Jealous Husbands
Three married couples would like to cross a river but have only a small canoe which can hold a maximum of two people at a time. However, all three husbands are insanely jealous and cannot bear to have their wife near another man while they are not around. Is it possible for all three couples to cross the river? If so, how? (10 points)
Gas in the Desert
You would like to drive your truck across a desert to a location 800 miles away. There are gas stations at each end of the 800 mile stretch but no gas stations anywhere in between. The truck can hold a maximum of 500 units of gasoline. Each unit of gas will allow the truck to travel one mile. You have several storage tanks which may be used to store gas at points in the desert for later use. Is it possible to cross the desert? If so, how many total units of gas are required?
The less gas required, the better the answer.
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Less than 1800 units of gas: |
25 points |
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Between 1800 and 1900 units of gas: |
20 points |
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Between 1900 and 2000 units of gas: |
15 points |
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Between 2000 and 2500 units of gas: |
10 points |
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More than 2500 units of gas: |
5 points |
BONUS QUESTION: Zeno’s Paradox
An ancient greek philosopher named Zeno came up with the following paradox. There is a race between legendary Greek hero Achilles and a tortoise. The tortoise is given a small head start. Zeno suggests that Achilles could never pass the tortoise:
Let A represent Achilles’ starting position and B represent the tortoise’s starting position. The race begins and it doesn’t take Achilles very long to reach point B. But, even though the tortoise is moving slowly, it certainly would have advanced from point B to a new point which we can call C. It doesn’t take Achilles long to reach point C, but once again in that time the tortoise would have moved forward a little bit to a new point which we can call D.
You can see that according to this line of reasoning, every time Achilles reaches the tortoise’s old position, the tortoise has moved on ahead to a new position. This seems to suggest that Achilles could never pass the tortoise. Where is the error in this reasoning? (10 points)
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