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. If n is a positive integer with n being even, then (𝑛3 − 𝑛)is even.
2. If n is odd, then n 2 is odd.
3. If n is an element of the integers and n is even, then 3𝑛 + 5 is odd.
4. The difference of the squares of any two consecutive integers is odd. Write in the If-Then
form and then either prove or disprove.
5. The sum of any three consecutive integers is divisible by three. Write in the If-Then form
and then either prove or disprove.
6. Proved by a deductive proof. If n is even, then 7n + 4 is even.
7. 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.
8. Give a deductive proof. If a and b are consecutive integers, then(𝑎 + 𝑏)2 is an odd number.
9. True or false. If true, provide a deductive proof. If false, provide a counter example.
If m and n are consecutive integers, then 4 divides (𝑚2 + 𝑛2 ).
11. x  y if and only if
2

x  y
xy 
4
a. Write the statement as 2 if-then statements.
12. 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
13. 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:
Evey 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.