Lesson 8: Inductive Logic, Conjectures, and Counterexamples

Inductive Logic, Conjectures, and Counterexamples

SCIENCE!!

​Inductive Reasoning

​Inductive reasoning finds patterns and uses those patterns to make conjectures regarding the future terms of the sequence. A conjecture is an unproven statement that is based on observations.

​Inductive Reasoning

We can use inductive reasoning to form conjectures in order to predict the future terms of specific sequences of patterns, numerical sequences, and even natural phenomena. Using the sequence to the right, find a pattern. How many blocks are added each time? How many blocks will there be in the 9th figure of the sequence?

​Inductive Reasoning

How many blocks are added each time? The number of blocks added each time is equal to the number of the figure in the sequence. Blocks are added at the bottom of each figure.

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How many blocks will there be in the 9th figure of the sequence?

​45 blocks.

Numerical Sequences

​We can also use inductive reasoning by observing patterns within numerical sequences and determining the future terms within that sequence. First, we must identify the constant change between each given term. This difference can be addition, subtraction, multiplication, division, or even exponential.

Numerical Sequences

In the sequence to the right, we need to figure out how 4 becomes 7. Well, we can add 3 to 4 and get 7:

​4 + 3 = 7

​How do we get 7 to 12? Add 5 to 7:

​7 + 5 = 12

​Now, does this pattern work for later terms as well?

​12 + 3 = 15

​15 + 5 = 20

Numerical Sequences

It seems that this pattern does work. So, for the first term, you add 3. For the next, you add 5. Then, you alternate between those two operations. Now, what is the next term in the sequence? What is the 8th term in the sequence?

Numerical Sequences

What is the next term in the sequence? 20 + 3 = 23

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What is the 8th term in the sequence? 23 + 5 = 28; 28 + 3 = 31

Multiple Choice

Two patterns are listed:

Pattern 1: 1, 2, 4, 8, 16

Pattern 2: 0, 3, 6, 9, 12

What are the next terms in both patterns?

• 20,14

• 32,15

• 18,15

• 24,14

Multiple Choice

Following the pattern, what is the next figure?

​Making a Conjecture

​We can make conjectures about any given terms that can form specific patterns. For example, I can make a conjecture about the product of any given even and odd integer: that they will give me an even result every time. I would probably phrase it like this: The product of any even and odd integer is an even integer.

​Testing a Conjecture

I then would need to test my conjecture using many small tests. For instance:

​2 x 3 = 6 (even product)

​4 x 3 = 12 (even product)

​-1 x 8 = -8 (even product)

​0 x 9 = 0 (even product)

For all four of these tests, my prediction is correct. So, my conjecture at least seems likely.

​Making a Conjecture

Let's practice making a conjecture. To the right, we are given an increasing number of collinear points. We are then tasked with connecting each point to each other point only once and count the number of connections we can make. We can predict the future number of connections by detecting a pattern in how many connections are being added each time.

​Making a Conjecture

​Notice that the number of connections added with each new point depends on how many points there are. Adding one point added one connection. Adding a third point added 2 more connections and so on (as indicated by the bottom row of the table). How many connections can be made in the 5th term? How about the 12th?

​Making a Conjecture

How many connections can be made in the 5th term? 10; the number of connections added increases by 1 with each new point added. Since 3 connections were added between the 3rd and 4th terms, 4 will be added between the 4th and 5th terms.

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How about the 12th? 66

​Counterexamples

​Many conjectures can be disproven by counterexamples. For example, if I make the conjecture that any product of three integers that includes at least one negative will be negative, then I can try to prove it using the following examples:

​-3 x 1 x 2 = -6 (negative result)

​-2 x -5 x -2 = -20 (negative result)

​Counterexamples

​However, given my conjecture that the products of any 3 integers which includes a negative will be negative, there are counterexamples, or examples which show my conjecture is false. One such counterexample is:

​-2 x -3 x 1 = 6 (positive result)

​Counterexamples

​It is important to remember that everyone has blind spots and may make mistakes. This means that we should examine conjectures with as much scrutiny as possible, attempting every possible counterexample until we are satisfied that the conjecture is solid.

​Practice!

​Find a counterexample to my initial conjecture that the difference of any two integers is smaller than both of the original integers.

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​Find a counterexample to the conjecture that the product of any two prime numbers is an odd number.

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​Find a counterexample to the conjecture that the sum of any two integers is greater than both of the original integers.

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​Find a counterexample to the conjecture that the least common multiple of any two whole numbers is smaller than both of those whole numbers.

Multiple Choice

To fully disprove a conjecture, one needs to find only ONE counterexample.

• True

• False

Multiple Choice

Which of the following is a counterexample to the following conjecture? If $$x^2\ =\ 4$$ , then x = 2

• x = 4

• x = -2

• x = 2

• x = -4

Multiple Choice

Which of the following conjectures is false?

• The product of two even numbers is even.
• The sum of two even numbers is even.
• The product of two odd numbers is odd.
• The sum of two odd numbers is odd.