Download Math 3: Unit 1 – Reasoning and Proof Inductive, Deductive

Math 3: Unit 1 – Reasoning and Proof
Inductive, Deductive
Quick Check on Lesson 1
Name:_______________________
Objective:
To use inductive reasoning to make conjectures. To use deductive reasoning to
prove conjectures.
1. Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence.
1
1
a) 384, 192, 96, 48, …
b)
2, 2 , 3, 3 , …
c) 1, 4, 9, 16, …
2
2
2. The price of overnight shipping was $8.00 in 2000, $9.50 in 2001, and $11.00 in 2002. Make
a conjecture about the price in 2003.
3. Find a counterexample for each conjecture.
a. Any number and its absolute value are opposites.
b. If a number is divisible by 5, then it is divisible by 10.
4. Identify the hypothesis and the conclusion of this conditional statement:
a) If the sky is blue, then it is sunny.
b) If y – 3 = 5, then y = 8.
5. Show that this conditional is false by finding a counterexample:
If the name of a state contains the word New, then the state borders an ocean.
6. Write the converse of the following conditional:
If two lines are not parallel and do not intersect, then they are skew.
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7. If n is an integer greater than 1 for which 2 n  1 is prime, then n is prime.
p:
q:
8. Shane made the following assertion: All numbers that are divisible by 4 are even numbers.
a. Write Shane’s assertion in if-then form.
b. Check Shane’s assertion by using inductive proof.
c. Give a deductive proof of Shane’s assertion.
9. Suppose it is true that “all members of the senior class are at least 5 feet 2 inches tall.” What,
if anything, can you conclude with certainty about each of the following students?
a. Darlene, who is a member of the senior class.
b. Trevor, who is 5 feet 10 inches tall
c. Anessa, who is 5 feet tall
d. Ashley, who is not a member of the senior class
10. In 1742, number theorist Christian Goldbach (1690 – 1764) wrote a letter to mathematician
Leonard Euler in which he proposed a conjecture that people are still trying to prove or
disporve. Goldbach’s Conjecture states:
Every even number greater than or equal to 4 can be expressed as the sum of 2 prime
numbers.
a. Verify Goldbach’s conjecture is true for 12 and 28.
b. Write Goldbach’s conjecture in if-then form.
c. Write the converse of Goldbach’s Conjecture. Is the converse true? Give a counterexample or
a deductive proof.
11. Determine whether the following are valid or invalid. Justify your reasoning.
a. If someone buys a new Lamborghini, she or he will pay over $200,000. Marie does not buy a
new Lamborghini. Therefore, Marie does not pay over $200,000 for her new car.
b. In China, job applicants do not ask how much they will be paid when they are hired. When
Jin Tai was hired, he asked his employer how much he would be paid. Jin Tai must have been
hired outside of China.
12. Use the true statements to determine whether the conclusion is true or false. Explain your
reasoning.
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


a.
If Diego goes shopping, then he will buy a pretzel.
If the mall is open, then Angela and Diego will go shopping.
If Angela goes shopping, then she will buy pizza.
The mall is open
Diego bought a pretzel.
b. Angela and Diego went shopping.
c. Angela bought a pretzel.
d. Diego had some of Angela’s pizza.
13. Given that the statement is of the form 𝑝 → 𝑞, write p and q. Then write the converse of
each statement.
a. If Jed gets a C on the exam, then he will get an A for the quarter.
p:
q:
Converse:
b. If the fuse has blown, then the light will not go on.
p:
q:
Converse:
Determine if each argument is valid or faulty.
14. 1) If you buy Tuff Cote luggage, it will survive airline travel.
2) Justin buys Tuff Cote luggage.
Conclusion: Justin’s luggage will survive airline travel.
15. 1) If you use Clear Line long distance service, you will have clear reception.
2) Anna has clear long distance reception.
Conclusion: Anna uses Clear Line long distance service.
Select the conclusion that makes the statement true.
16. All equilateral triangles are isosceles. ΔABC is equilateral. Therefore ΔABC (is, is not, or
may be) isosceles.
17. The Oak Terrace apartment building does not allow dogs. Serena lives at Oak Terrace. So,
Serena (must, may, may not) keep her dog.
18. The Kolob Arch is the world’s widest natural arch. The world’s widest arch is in Zion
National Park. So, the Kolob Arch (is, may be, is not) in Zion.
19. Zion National Park is in Utah. Jeremy spent a week in Utah. So, Jeremy (must have, may
have, never) visited Zion National Park.