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Geometry

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Day 10 Today s Agenda Inductive reasoning Counterexamples Conditional Statements Inverse Converse Contrapositive Truth Tables Conjunctions Disjunctions ... – PowerPoint PPT presentation

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Title: 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
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