Conjectures and Counterexamples
___________ is the process of reaching a
conclusion based on a pattern of examples.
Inductive reasoning
A ________ is any sentence that is either true T or false F, but not both.
Statement
An argument is _____ if it is impossible for all
the premises, or supporting statements.
valid
What are supplementary angles?
two angles with measures
that have a sum of 180°.
What are complementary angles?
two angles with measures that
have a sum of 90°.
You can use inductive reasoning to make an educated guess based on known information and specific examples. This educated guess is also known as a?
Conjecture
What the truth or falsity of a statement?
Truth Value
Is this Deductive or Inductive?
Newton’s first law of motion states that an object at rest will remain at rest unless acted on by an unbalanced force. Elisa watches a soccer ball roll across the field. She concludes that an unbalanced force has acted upon the soccer ball.
Deductive
Lines, segments, or rays that intersect at right angles are?
perpendicular
Statements, Conditionals, and Biconditionals:
A compound statement using the word
and is called a?
Conjunction
What are the patterns?
Appointment times: 8:30 A.M., 9:15 A.M., 10:00 A.M.,
10:45 A.M., . . .
+45 minutes
The _______ of a statement has the opposite
meaning, as well as the opposite truth value, of the original statement.
negation
Unlike inductive reasoning, which uses a specific pattern of examples or observations to make a general conclusion, __________ uses general facts, rules, definitions, or properties to reach specific valid conclusions from given statements.
deductive reasoning
The difference between the measures of two supplementary angles is 18°. The measure of the smaller angle is ____, and the measure of the larger angle is ____.
81° and 99°
It only takes one example that contradicts the
conjecture, however, to show that a conjecture is not
always true. This example is called a?
Counterexample
The sum of two odd numbers is always an _____ number.
Even
Two or more statements joined by the word and or or form a...
compound statement
Every time Tamika has worn her favorite jersey to a
football game, her school’s team has won the game.
Tamika is wearing her favorite jersey to the football
game tonight, so she concludes that her school’s
team will win the game.
Tamika is basing her conclusion on a specific pattern of observations, so she is using inductive reasoning.
Certain relationships can be assumed from a figure, but most cannot.
Interpreting Diagrams
Deductive Reasoning:
Is this Deductive or Inductive?
Mrs. Jackson notices that her family’s data usage is increasing by approximately 2500 megabytes of data every month. So, she concludes that her family’s data usage next month will be 2500 megabytes greater than this month’s data usage.
Inductive
The intensity of a sound to the human ear is measured in A-weighted decibels, or dBA. For every 3 decibels over 85 decibels, the exposure time it takes to cause hearing damage is cut in half. How long does it take to cause hearing damage at 106 decibels? Write your answer as a decimal.
3.75 minutes
A compound statement using the word or is called a?
disjunction
Scientists have found that the quality and amount of
sleep greatly impact learning and memory. Lack of sleep causes students to have trouble focusing and receiving new information. Sleep deprivation also makes it difficult to retrieve previously learned information. Draw a valid conclusion from the given statements, if possible.
Given: If you are tired, then you will not do well on your test. If you do not get enough sleep, then you will be tired.
The measures of the angles are 17°and 73° are the Complementary or not?
Yes; 17° + 73° = 90°, so the two angles are complementary.
A ___________ is a compound statement that
consists of a premise, or hypothesis, and a conclusion,
which is false only when its premise is true and its
conclusion is false.
conditional statement