Inductive Reasoning and Conjecture

  1. Show how to make conjectures from patterns.

The table shows the total sales for the first three months a store is open. The owner wants to predict the sales for the fourth month.

Month / Sales
1 / $500
2 / $1500
3 / $4500

Make a statistical graph that best displays the data.

Make a conjecture about the sales in the fourth month and justify your claim or prediction.

The sales triple each month, so in the fourth month there will be $13,500 in sales.

  1. Show how to make a conjecture about a single figure

Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture.

The sum of an odd number and an even number,

The sum of an odd number and an even number is odd; 3 + 4 = 7, 5 + 10 = 15.

For points L, M, and N, LM = 20, MN = 6, and LN = 14. Make a conjecture and draw a figure to illustrate your conjecture.

L, M, and N are collinear.

Inductive Reasoning and Conjecture

  1. Show how to find a counterexample based on given information.

Based on the table showing unemployment rates for various counties in Texas, find a counterexample for the following statement.

The unemployment rate is highest in the cities with the most people.

Maverick has only 50,436 people in its population, and it has a higher rate of unemployment than El Paso, which has 713,126 people in its population.

  1. Determine Truth Values

show how to find the truth value of conjunctions and disjunctions of sentences.

  1. Use the following statements to write a compound statement for each conjunction. Then find its truth-value. Explain your reasoning.

p: One foot is 14 inches.

q: September has 30 days.

r: A plane is defined by three non-collinear points.

  1. p and q One foot is 14 inches, and September has 30 days;

Although q is true, p is false. So p and q is false.

b. ∼p∧r A foot is not 14 inches, and a plane is defined by three noncollinear points; Both ∼p and r are true, so ∼p∧ r is true.

  1. Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.

p: AB is proper notation for “segment AB.”

q: Centimeters are metric units.

r: 9 is a prime number.

  1. p or q AB is proper notation for “segment AB,” or centimeters are metric units; Both p and q are true, so p or q is true.

b.q∨ r Centimeters are metric units, or 9 is a prime number; Since q is true, q∨r is true.

c.∼p∨ r Since both ∼p and r are false, ∼p ∨ r is false.

  1. Construct a truth table for each compound statement.
  1. ~p V q

p / q / ~p / ~p V q
T / T / F / T
T / F / F / F
F / T / T / T
F / F / T / T
  1. p∨ (∼q∧r )

p / q / r / ~q / (∼q∧r ) / p∨ (∼q∧r )
T / T / T / F / F / T
T / T / F / F / F / T
T / F / T / T / T / T
T / F / F / T / F / T
F / T / T / F / F / F
F / T / F / F / F / F
F / F / T / T / T / T
F / F / F / T / F / F
  1. Venn Diagrams

Show how to use a Venn diagram to make conjectures.

Students should be able to make a conjecture, write a compound statement, and find its truth-value.

  1. The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes.
  1. How many students are enrolled in all three classes? 9
  2. How many students are enrolled in tap or ballet? 121
  3. How many students are enrolled in jazz and ballet, but not tap? 25

Venn Diagram

Truth tables can also be helpful in evaluating the truth values of statements.

◦Truth Tables

p / q / p∧q / p / q / p∨q
T / T / T / T / T / T
F / T / F / T / F / T
F / F / F / F / T / T
T / F / F / F / F / F
A conjunction is true only when p and q are true. / A disjunction is false only when both p and q are false.

In a negation, if p is true, then ∼p is false. If p is false, then ∼p is true.