2.1 Inductive Reasoning

ReviewGeometry R/H

Inductive ReasoningWhen you make a prediction based on several

examples, you are applying inductive reasoning.Formal Definition: Inductive reasoning is the

process of reasoning that a rule or statement is true because specific cases (examples that you observe) are true.

Making a ConjectureA statement you believe to be true based on

inductive reasoning is called a conjecture.Example 2: Complete the conjecture. The sum of two positive numbers is ? List some examples and look for a pattern.

1 + 1 = 2 3.14 + 0.01 = 3.15 3,900 + 1,000,017 = 1,003,917

The sum of two positive numbers is positive.

CounterexamplesTo show that a conjecture is true you must

prove it, but to show a conjecture is false, you need only to find one example for which the conjecture is false. This is called a counterexample.

In the following conjecture, show that it is not true by finding a counterexample.

For any real number x, x 2 ≥ x A counterexample could be x = ½ x 2 = (½)2 = ¼ ¼ ≥ ½ therefore, the conjecture is not true.

Steps in Inductive Reasoning

Step 1: Look for a Pattern

Step 2: Make a Conjecture

Step 3:Is it True?Verify the

ConjectureFind a

Counter-example

yes no

Find the next item in each pattern.1. 0.7, 0.07, 0.007, … 2.

Review of 2.1

0.0007Determine if each conjecture is true. If false, give a counterexample.3. The quotient of two negative numbers is a positive number.4. Every prime number is odd.5. Two supplementary angles are not congruent.

6. The square of an odd integer is odd.

false; 2true

false; 90° and 90°true

Go over Sequences PacketArithmetic, Quadratic, Triangular, Geometric Sequences

** (NOT in Book)

2.2 Logic and Venn Diagrams

Geometry R/H

Conjunction and Disjunction

Let p: I am tired.Let q: I will fall asleep

Now you try!Let p: Let q:

A Venn Diagram is a diagram made of inter-locking circles that we can use to solve a logic problem.

Let’s try solving the following problem using a Venn Diagram:

In a class of 50 students, 18 take Chorus, 26 take Band, and 2 take both Chorus and Band. How many students in the class are not enrolled in either Chorus or Band?

What is a Venn Diagram?

Here’s the Venn Diagram:First we start with a rectangle to represent

the whole class of 50Add a circle for the students in Chorus (18

people will be in this circle.) Add a circle for the

students taking Band (26 people will be in this circle.)

Chorus

Band

Total = 50

How do we solve the problem?Start with the number of students taking both

classes – that will have to be inside both circles.Since we now have 2 people inside the circle of

chorus, how many more do we need to add to

How many do we need to add to have a total of 26 in band?

make a total of 18?

Chorus

Band2 2416

Total = 50

How do we solve the problem?We now have a total of 16 + 2 + 24 or 42

students that are either taking band or chorus. There are 50 students in the class, so that

means that 8 people are not enrolled in either band or chorus.

We place those 8 people in this area.

8

Chorus

Band2 2416

Total = 50

1918 55Chocolate

Vanilla

12

We add all the numbers together to get the total number of people in the survey.That would be?

A survey is taken at an ice cream parlor. People are asked to list their two favorite flavors. 74 list vanilla as one of their favorite flavors while 37 list chocolate. If 19 list both flavors and 12 list neither of these two flavors, how many people participated in the survey?

104

That is 25. But how can we

divide 25 into the two circles?

Thirty students went to lunch. Pizza and ham-burgers were offered. 15 students chose pizza, and 13 chose hamburgers. 5 students chose neither. How many students chose only hamburgers?

If we subtract 5 from 30 we will get the number of people that should be in one of the circles.

5

Pizza Burgers

Total = 30

If we add the 15 people who wanted pizza to the 13 who wanted hamburgers, we get 28.

This is 3 more than the total of 25 that we should have in one of the circles

This means that those 3 people are being counted twice and so is the number that belongs in the center.

5

Pizza Burgers

Total = 30 Finish the diagram, and

answer the question. The students that

chose only hamburgers is 10.

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