Unit+1+Lesson+1+Inductive+&+Deductive+Reasoning


 * Inductive & Deductive Reasoning**


 * Inductive Reasoning:** reasoning from patterns based on the analysis of specific cases. You can NOT prove a conjecture using algebraic reasoning. Often times you will use inductive reasoning to come up with a conjecture.


 * Conjecture:** In mathematics, a conjecture is a mathematical statement, which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic.

Ex: What is true about the sum of two odd integers?

If you were using inductive reasoning you would add some pairs of odd integers and look for a pattern.

1 + 3 = 4 3 + 5 = 8 5 + 7 = 12 7 + 9 = 16 9 + 11 = 20 11 + 13 = 24

All of the sums appear to be even.

So a possible conjecture would be: If you add two odd integers, then the sum will be an even integer.

➢ Remember this is only a conjecture, by trying six cases or even trying a thousand cases does not prove that this will be true for every two odd integers.


 * Deductive Reasoning:** reasoning from facts, definitions, and accepted properties to new statements using principles of logic. By using correct deductive reasoning the conclusions you reach are certain. You can prove a conjectures using deductive reasoning.

Note: You should know the following relationships:
 * Even Integers** can be written in the form: 2m, where m is an integer. (Even numbers have a factor of two.)
 * Odd Integers** can be written in the form: 2n + 1, where n is an integer. (One more than an even number will always be an odd.)
 * Consecutive Integers** can be written in the form: n, n + 1, n + 2, etc, where n is an integer. (By adding one more to the previous number you will get the next consecutive integer.)
 * Consecutive Odd Integers** can be written in the form: 2n + 1, 2n + 3, 2n + 5, etc, where n is an integer. (By adding two more to the previous number you will get the next consecutive odd integer.)
 * Consecutive Even Integers** can be written in the form: 2n, 2n + 2, 2n + 4, etc, where n is an integer. (By adding two more to the previous number you will get the next consecutive even integer.)


 * Write a deductive proof that proves that the sum of two odd integers is even.**

Let a and b be two odd integers such that a = 2n + 1 and b = 2m + 1, where m and n are integers. (You need to pick different variables when defining a and b, otherwise they would represent the same odd integer and we want to show that this relationship is true for any two odd integers.) If you add a and b then you will get, which is an even integer. Therefore, if you add two odd integers then the sum will be even.


 * Ex: What happens when you multiply two even integers?**

2 x 4 = 8 6 x 4 = 24 2 x 10 = 20 4 x 12 = 48 2 x 6 =12
 * Use inductive reasoning to form a conjecture (Try some cases):**


 * Conjecture**: If you multiply two even integers then the product will be even.

Three Steps: 1. Define the variables. 2. Use algebra to prove the conjecture. 3. State what you have proven.
 * Now use deductive reasoning to prove your conjecture:**

Let a and b be two even integers such that a = 2n and b = 2m, where m and n are integers. ab = (2n)(2m)=2(2mn) Therefore, if you multiply two even integers then the product will be even.


 * By showing the product has a factor of 2 you are proving that it is even.*


 * Ex: What happens when you multiply two odd integers?**

1 x 3 = 3 5 x 3 = 15 7 x 3 = 21 9 x 3 = 27 11 x 5 = 55 1 x 7 =7
 * Use inductive reasoning to form a conjecture (Try some cases):**


 * Conjecture:** If you multiply two odd integers then the product will be odd.

Let a and b be two odd integers such that a = 2n + 1 and b = 2m + 1, where m and n are integers. ab = (2n + 1)(2m+ 1)=(4mn + 2n + 2m + 1) = 2(2mn + n + m) +1. Therefore, if you multiply two odd integers then the product will be odd.
 * Now use deductive reasoning to prove your conjecture:**


 * By showing the product is 2 times an integer plus 1 you are proving that it is odd.*