Geometry

1
Geometry

• Day 10

2
Todays Agenda

• Inductive reasoning
• Counterexamples
• Conditional Statements
• Inverse
• Converse
• Contrapositive
• Truth Tables
• Conjunctions
• Disjunctions
• Biconditionals
• Venn diagrams

3
Standards

• Make conjectures with justifications about
  geometric ideas. Distinguish between information
  that supports a conjecture and the proof of a
  conjecture.
• Find the converse, inverse, and contrapositive of
  a statement.
• Use truth tables to determine the truth values of
  propositional statements.

4
Inductive Reasoning

• Inductive reasoning is the process of using
  examples and observations to reach a conclusion.
• Any time you use a pattern to predict what will
  come next, you are using inductive reasoning.
• A conclusion based on inductive reasoning is
  called a conjecture.
• Turn to p. 90 and complete Guided Practice
  problems 1-2.

5
Counterexamples

• A conjecture is either true all of the time, or
  it is false.
• If we wish to demonstrate that a conjecture is
  true all the time, we need to prove it through
  deductive reasoning.
• We will have more on deductive reasoning and the
  proof process later. But for now, know that we
  can never prove an idea by offering examples that
  support the idea.
• However, it can be easy to demonstrate that a
  conjecture is false. We simply need to provide a
  counterexample.
• P. 92, Guided Practice 4

6
Practice

• Complete 48 on page 95.
• Now complete 50.
• Careful! Inductive reasoning is only as good as
  our observations. If we encounter new data that
  contradicts our conjecture, we need to revise the
  conjecture.

7
Intro to Logic

• A statement is a sentence that is either true or
  false (its truth value).
• Logically speaking, a statement is either true or
  false. What are the values of these statements?
• The sun is hot.
• The moon is made of cheese.
• A triangle has three sides.
• The area of a circle is 2pr.
• Statements can be joined together in various ways
  to make new statements.

8
Conditional Statements

• A conditional (or propositional) statement has
  two parts
• A hypothesis (or condition, or premise)
• A conclusion (or result)
• Many conditional statements are in If then
  form.
• Ex. If it is raining outside, then I will get
  wet.
• A conditional statement is made of two separate
  statements each part has a truth value. But the
  overall statement has a separate truth value.
  What are the values of the following statements?
• If today is Friday, then tomorrow is Saturday.
• If the sun explodes, then we can live on the
  moon.
• If a figure has four sides, then it is a square.

9
Conditional Statements

• Conditional statements dont have to be If
  then See if you can determine the condition
  and conclusion in each of the following, and
  restate in If then form.
• An apple a day keeps the doctor away.
• What goes up must come down.
• All dogs go to heaven.
• Triangles have three sides.

10
Inverse

• The inverse of a statement is formed by negating
  both its premise and conclusion.
• Statement
• If I take out my cell phone, then Mr. Peterson
  will confiscate it.
• Inverse
• If I do take out my cell phone, then Mr.
  Peterson will confiscate it.

not
not
11
Try these

• Give the inverses for the following statements.
  (You may wish to rewrite as If then first.)
  Then determine the truth value of the inverse.
• Barking dogs give me a headache.
• If lines are parallel, they will not intersect.
• I can use the Pythagorean Theorem on right
  triangles.
• A square is a four-sided figure.

12
Converse

• A statements converse will switch its hypothesis
  and conclusion.
• Statement
• If I am happy, then I smile.
• Converse
• If , then .

I smile
I am happy
13
Try these

• Give the converses for the following statements.
  Then determine the truth value of the converse.
• If I am a horse, then I have four legs.
• When Im thirsty, I drink water.
• All rectangles have four right angles.
• If a triangle is isosceles, then two of its sides
  are the same.

14
Contrapositive

• A contrapositive is a combination of a converse
  and an inverse. The premise and conclusion
  switch, and both are negated.
• Statement
• If my alarm has gone off,then I am awake.
• Contrapositive
• If
  ,then .

my alarm has not gone off
not

I am not awake
not

15
Try these

• Give the contrapositives for the following
  statements. Then determine its truth value.
• If it quacks, then it is a duck.
• When Superman touches kryptonite, he gets sick.
• If two figures are congruent, they have the same
  shape and size.
• A pentagon has five sides.
• Note A contrapositive always has the same truth
  value as the original statement!

16
Symbolic representation

• Logic is an area of study, related to math (and
  computer science and other fields). In formal
  logic, we can represent statements symbolically
  (using symbols).
• Some common symbols
• a statement, usually a premise a statement,
  usually a conclusion creates a conditional
  statement negates a statement (takes its
  opposite)

17
Examples

• If p, then q
• InverseIf not p, then not q
• ConverseIf q, then p
• ContrapositiveIf not q, then not p

18
Truth Table

• A truth table is a way to organize the truth
  values of various statements.
• In a truth table, the columns are statements and
  the rows are possible scenarios.
• The table contains every possible scenario and
  the truth values that would occur.
• Example

T
F
T
F
19
A conditional truth table
T
T
T
T
F
F
F
T
T
F
F
T
20
A conditional truth table
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
T
T
T
T
21
Logical Equivalents

• Two statements are considered logical equivalents
  if they have the same truth value in all
  scenarios. A way to determine this is if all the
  values are the same in every row in a truth table.

22
Logical Equivalents

• Which of the following statements are logically
  equivalent?

T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
T
T
T
T
23
Conjunctions

• A conjunction consists of two statements
  connected by and.
• Example
• Water is wet and the sky is blue.
• Notation
• A conjunction of p and q is written as

24
Conjunctions

• A conjunction is true only if both statements are
  true.

• Remember the truth value of a conjunction refers
  to the statement as a whole.
• Consider The sun is out and it is raining.

T
T
T
T
F
F
T
F
F
F
F
F
25
Disjunctions

• A disjunction consists of two statements
  connected by or.
• Example
• I can study or I can watch TV.
• Notation
• A disjunction of p and q is written as

26
Disjunctions

• A disjunction is true if either statement is true.

• Consider Timmy goes to Stanton or he goes to
  Paxon.

T
T
T
T
F
T
T
F
T
F
F
F
27
Biconditional

• A biconditional statement is a special type of
  conditional statement. It is formed by the
  conjunction of a statement and its converse.
• Example
• If a quadrilateral has four right angles then it
  is a rectangle, and if a quadrilateral is a
  rectangle then it has four right angles.
• Biconditional statements can be shortened by
  using if and only if (iff.).
• A quadrilateral is a rectangle if and only if it
  has four right angles.
• This is true whether you read it forwards or
  backwards.

28
Biconditional

• A good definition will consist of a biconditional
  statement.
• Ex A figure is a triangle if and only if it has
  three sides.

29
Biconditional

• A biconditional is true when the statements have
  the same truth value.

• Consider Two distinct coplanar lines are
  parallel if and only if they have the same
  slope.
• Our team will win the playoffs if and only if
  pigs fly.

T
T
T
T
F
F
T
F
F
F
F
T
30
Venn Diagrams

• The truth values of compound statements can also
  be represented in Venn diagrams.
• p A figure is a quadrilateral.
• q A figure is convex.
• Which part of the diagramrepresents

p
q
31
Venn Diagrams Conditionals

• A Venn diagram can represent a conditional
  statement
• p A figure is a quadrilateral.
• q A figure is a square.

p
q
32
Can you?

• Use inductive reasoning to recognize patterns and
  make predictions?
• Give a counter-example to disprove a conjecture?
• Identify the hypothesis and conclusion of a
  conditional statement?
• Write the converse, inverse, and contrapositive
  of a conditional statement?
• Create a truth table to examine scenarios?
• Recognize conjunctions, disjunctions, and
  biconditional statements?
• Evaluate logic using Venn diagrams?

33
Assignments

• Homework 5
• Wkbk, pp. 15, 19
• Homework 6
• Truth Tables Handout
• Textbook
• pp. 102-103 31, 33, 41-47
• pp. 112 59-61