Welcome to Interactive Chalkboard

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Welcome to Interactive Chalkboard
2.2 Logic
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Objectives

• Determine truth values of conjunctions and
  disjunctions
• Construct truth tables
• Construct and interpret Venn Diagrams

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Truth Values

• A statement is any sentence that is either true
  or false, but not both. The truth or falsity of a
  statement is its truth value.
• Statements are most often represented using a
  letter such as p or q.
• Examplep Denver is the capital of
  Colorado.

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Truth Values

• The negation of a statement has the opposite
  meaning as well as the opposite truth value of
  the original statement.
• Example (using the previous statement)
• not p also written as p Denver is not
  the capital of Colorado.

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Truth Values

• A compound statement is the joining of two or
  more statements.
• Example p Denver is a city in Colorado.q
  Denver is the capital of Colorado.p and q
  Denver is a city in Colorado, and
  Denver is the capital of
  Colorado.

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Truth Values

• When we join two statements with the word and
  as in the previous example we have created a
  conjunction.
• We write conjunctions as p q, which is read as
  p and q.
• A conjunction is true only when BOTH statements
  in it are true.

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Example 1a
Use the following statements to write a compound
statement for the conjunction p and q. Then find
its truth value.p One foot is 14 inches.q
September has 30 days.r A plane is defined by
three noncollinear points.
Answer One foot is 14 inches, and September has
30 days. p and q is false, because p is false
and q is true.
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Example 1b
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Example 1c
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Example 1d
Answer A foot is not 14 inches, and a plane is
defined by three noncollinear points. p ? r is
true, because p is true and r is true.
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Your Turn
Answer June is the sixth month of the year, and
a turtle is a bird false.
Answer A square does not have five sides, and a
turtle is not a bird true.
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Your Turn
Answer A square does not have five sides, and
June is the sixth month of the year true.
Answer A turtle is not a bird, and a square has
five sides false.
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More About Truth Values

• Statements can also be joined by the word or.
  We call these disjunctions and write them as p
  V q, which is read as p or q.
• Example p Susan has 1st lunch.
• q Susan has 2nd lunch.
• p V q Susan has 1st lunch, or Susan has
  2nd lunch.

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More About Truth Values

• A disjunction is true if at least one of the
  statements is true. The truth value of a
  disjunction is only false if both of the
  statements are false.

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Example 2a
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Example 2b
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Your Turn
Answer 6 is an even number, or a triangle as 3
sides true.
v
Answer A cow does not have 12 legs, or a
triangle does not have 3 sides true.
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Venn Diagrams

• Often we illustrate conjunctions and disjunctions
  by using Venn Diagrams.
• The Venn Diagram to the right represents the
  number of students enrolled in each of the
  electives.

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Venn Diagrams
In a Venn Diagram the conjunction is represented
by the intersection of all sets, (i.e. the white
section of 9 students).
Meanwhile, a disjunction is simply represented by
the union of all the sets, (i.e. all of the
circles and intersections).
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Example 3
DANCING The Venn diagram shows the number of
students enrolled in Moniques Dance School for
tap, jazz, and ballet classes.
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Example 3a
How many students are enrolled in all three
classes?
The students that are enrolled in all three
classes are represented by the intersection of
all three sets.
Answer There are 9 students enrolled in all
three classes
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Example 3b
How many students are enrolled in tap or ballet?
The students that are enrolled in tap or ballet
are represented by the union of these two sets.
Answer There are 28 13 9 17 25 29 or
121 students enrolled in tap or ballet.
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Example 3c
How many students are enrolled in jazz and ballet
and not tap?
The students that are enrolled in jazz and ballet
and not tap are represented by the intersection
of jazz and ballet minus any students enrolled in
tap.
Answer There are 25 students enrolled in jazz
and ballet and not tap.
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Your Turn
PETS The Venn diagram shows the number of
students at Mustang Mid-High that have dogs,
cats, and birds as household pets.
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Your Turn
a. How many students in Mustang Mid-High have at
least one of three types of pets? b. How many
students have dogs or cats? c. How many
students have dogs, cats, and birds as pets?
Answer 311
Answer 280
Answer 10
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Truth Tables

• Last, a convenient method for organizing truth
  values of statements is to use truth tables.

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Truth Tables

• By constructing truth tables, you can organize
  the truth values for statement (p), its
  negation( p), any conjunctions of the statement
  (p q), any disjunctions of the statement (p v
  q), and even any negations of conjunctions ( p
  q ) or any negations of disjunctions ( p v
  q).

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Example 4a
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Example 4a
Step 2 List the possible combinations of truth
values for p and q.
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Example 4a
Step 3 Use the truth values of p to determine
the truth values of p.
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Example 4a
Step 4 Use the truth values for p and q to
write the truth values for p ? q.
Answer
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Example 4b
Step 1 Make columns with the headings p, q, r,
q, q ? r, and p ? (q ? r).
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Example 4b
Step 2 List the possible combinations of truth
values for p, q, and r.
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Example 4b
Step 3 Use the truth values of q to determine
the truth values of q.
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Example 4b
Step 4 Use the truth values for q and r to
write the truth values for q ? r.
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Example 4b
Step 5 Use the truth values for p and q ? r to
write the truth values for p ? (q ? r).
Answer
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Example 4c
Construct a truth table for (p ? q) ? r.
Step 1 Make columns with the headings p, q, r,
r, p ? q, and (p ? q) ? r.
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Example 4c
Construct a truth table for (p ? q) ? r.
Step 2 List the possible combinations of truth
values for p, q, and r.
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Example 4c
Construct a truth table for (p ? q) ? r.
Step 3 Use the truth values of r to determine
the truth values of r.
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Example 4c
Construct a truth table for (p ? q) ? r.
Step 4 Use the truth values for p and q to write
the truth values for p ? q.
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Example 4c
Construct a truth table for (p ? q) ? r.
Step 5 Use the truth values for p ? q and r to
write the truth values for (p ? q) ? r.
Answer
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Your Turn
Construct a truth table for the following
compound statement.
Answer
43
Your Turn
Construct a truth table for the following
compound statement.
Answer
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Your Turn
Construct a truth table for the following
compound statement.
Answer
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Assignment

• Geometry
• Pg. 72 73
• 18 32, 34, 36, 38, 41 44
• Pre-AP Geometry Pg. 72 73
• 18 32, 34, 36, 38, 40, 41 44, 51 and 52