welcome to interactive chalkboard 2.2 logic. objectives determine truth values of conjunctions and...

2.2 Logic2.2 Logic

ObjectivesObjectives

Determine truth values of Determine truth values of conjunctions and disjunctionsconjunctions and disjunctions

Construct truth tablesConstruct truth tables

Construct and interpret Venn Construct and interpret Venn DiagramsDiagrams

Truth ValuesTruth Values

A A statementstatement is any sentence that is is any sentence that is either true or false, but not both. The either true or false, but not both. The truth or falsity of a statement is its truth or falsity of a statement is its truth truth valuevalue. .

Statements are most often represented Statements are most often represented using a letter such as using a letter such as pp or or qq. .

Example:Example:pp: Denver is the capital of Colorado. : Denver is the capital of Colorado.


The The negation negation of a statement has the of a statement has the opposite meaning as well as the opposite meaning as well as the opposite truth value of the original opposite truth value of the original statement. statement.

Example Example (using the previous statement):(using the previous statement):

not p also written as ~ pnot p also written as ~ p: : Denver is not the capital of Denver is not the capital of

Colorado. Colorado.


A A compound statementcompound statement is the joining is the joining of two or more statements.of two or more statements.

Example: Example: pp: Denver is a city in Colorado.: Denver is a city in Colorado.qq: Denver is the capital of Colorado.: Denver is the capital of Colorado.

p and qp and q: Denver is a city in Colorado, : Denver is a city in Colorado, and Denver is the and Denver is the capital of capital of Colorado. Colorado.


When we join two statements with When we join two statements with the word “and” as in the previous the word “and” as in the previous example we have created a example we have created a conjunctionconjunction. .

We write conjunctions as We write conjunctions as p ^ qp ^ q, , which is read as “which is read as “p and qp and q.” .”

A conjunction is true only when A conjunction is true only when BOTH statements in it are true.BOTH statements in it are true.

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.

Example 1a:Example 1a:

Use the following statements to write a compound statement for the conjunction . 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: A plane is defined by three noncollinear points, and one foot is 14 inches. is false, because r is true and p is false.

Example 1b:Example 1b:

Use the following statements to write a compound statement for the conjunction . 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: September does not have 30 days, and a plane is defined by three noncollinear points.

is false because is false and r is true.

Example 1c:Example 1c:

Use the following statements to write a compound statement for the conjunction p r. 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: 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.

Example 1d:Example 1d:

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.

Use the following statements to write a compound statement for each conjunction. Then find its truth value.p: June is the sixth month of the year.q: A square has five sides.r: A turtle is a bird.

a. p and r

b.

Your Turn: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.

c.

d.

Use the following statements to write a compound statement for each conjunction. Then find its truth value.p: June is the sixth month of the year.q: A square has five sides.r: A turtle is a bird.


More About Truth ValuesMore About Truth Values

Statements can also be joined by the Statements can also be joined by the word “or.” We call these word “or.” We call these disjunctionsdisjunctions and write them as and write them as p p VV qq, which is read as “p or q.” , which is read as “p or q.”

Example:Example: p: Susan has 1 p: Susan has 1stst lunch. lunch. q: Susan has 2q: Susan has 2ndnd lunch. lunch.

p p VV q : Susan has 1q : Susan has 1stst lunch, or Susan lunch, or Susan has has 2 2ndnd lunch. lunch.

More About Truth ValuesMore About Truth Values

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

Use the following statements to write a compound statement for the disjunction p or q. Then find its truth value.

p: is proper notation for “line AB.”q: Centimeters are metric units.r: 9 is a prime number.

Answer: is proper notation for “line AB,” or centimeters are metric units. p or q is true because q is true. It does not matter that p is false.


Answer: Centimeters are metric units, or 9 is a prime number. is true because q is true. It does not matter that r is false.

Use the following statements to write a compound statement for the disjunction . Then find its truth value.

p: is proper notation for “line AB.”q: Centimeters are metric units.r: 9 is a prime number.


Answer: 6 is an even number, or a triangle as 3 sides; true.

Answer: A cow does not have 12 legs, or a triangle does not have 3 sides; true.

Use the following statements to write a compound statement for each disjunction. Then find its truth value.p: 6 is an even number.q: A cow has 12 legsr: A triangle has 3 sides.

a. p or r

b. v


Venn DiagramsVenn Diagrams

Often we illustrate Often we illustrate conjunctions and conjunctions and disjunctions by using disjunctions by using Venn DiagramsVenn Diagrams..

The Venn Diagram to The Venn Diagram to the right represents the right represents the number of the number of students enrolled in students enrolled in each of the electives. each of the electives.

Venn DiagramsVenn 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).

DANCING The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes.

Example 3:Example 3:

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


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.


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.


PETS The Venn diagram shows the number of students at Mustang Mid-High that have dogs, cats, and birds as household pets.


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


Truth TablesTruth Tables

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

~~p p ~~ppqqpp

Truth TablesTruth Tables

By constructing truth tables, you can By constructing truth tables, you can organize the truth values for organize the truth values for statement (p), its negationstatement (p), its negation(~ p), any conjunctions of the (~ p), any conjunctions of the statement (p ^ q), any disjunctions of statement (p ^ q), any disjunctions of

the statement (p the statement (p v q), and even any q), and even any negations of conjunctions (~ p ^ ~ negations of conjunctions (~ p ^ ~ q ) or any negations of disjunctions q ) or any negations of disjunctions

(~ p (~ p v ~~ q).q).

Step 1 Make columns with the headingsp, q, ~p, and ~p

Construct a truth table for .

~~p p ~~ppqqpp


Step 2 List the possible combinations of truth values for p and q.


FFFFTTFFFFTTTTTT

~~p p ~~ppqqpp


Step 3 Use the truth values of p to determine the truth values of ~p.


TTFFFFTTTTFFFFFFTTFFTTTT

~~p p ~~ppqqpp


Step 4 Use the truth values for ~p and q to write the truth values for ~p q.

Answer:


TTTTFFFFTTTTTTFFFFFFFFTTTTFFTTTT

~~p p ~~ppqqpp


pp ( (~~qq rr))~~qq rr~q~qrrqqpp

Step 1 Make columns with the headingsp, q, r, ~q, ~q r, and p (~q r).



Step 2 List the possible combinations of truth values for p, q, and r.


FFFFFF

pp ( (~~qq rr))~~qq rr~q~q

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp


Step 3 Use the truth values of q to determine the truth values of ~q.


TTFFFFFF

pp ( (~~qq rr))~~qq rr

FF

TT

FF

TT

FF

TT

FF

~q~q

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp


Step 4 Use the truth values for ~q and r to write the truth values for ~q r.


FFTTFFFFFF

pp ( (~~qq rr))

FF

TT

FF

FF

FF

TT

FF

~~qq rr

FF

TT

FF

TT

FF

TT

FF

~q~q

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp


Step 5 Use the truth values for p and ~q r to write the truth values for p (~q r).

Answer:


FFFFTTFFFFFF

FF

TT

FF

TT

TT

TT

TT

pp ( (~~qq rr))

FF

TT

FF

FF

FF

TT

FF

~~qq rr

FF

TT

FF

TT

FF

TT

FF

~q~q

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp


Construct a truth table for (p q) ~r.

((pp qq) ) ~~rrpp qq~r~rrrqqpp

Step 1 Make columns with the headingsp, q, r, ~r, p q, and (p q) ~r.


Step 2 List the possible combinations of truth values for p, q, and r.

FFFFFF

((pp qq) ) ~~rrpp qq~r~r

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp



Step 3 Use the truth values of r to determine the truth values of ~r.


TTFFFFFF

((pp qq) ) ~~rrpp qq

TT

FF

FF

TT

TT

FF

FF

~r~r

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp


Step 4 Use the truth values for p and q to write the truth values for p q.


FFTTFFFFFF

((pp qq) ) ~~rr

TT

FF

TT

TT

TT

TT

TT

pp qq

TT

FF

FF

TT

TT

FF

FF

~r~r

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp


Step 5 Use the truth values for p q and ~r to write the truth values for (p q) ~r.

Answer:


FFFFTTFFFFFF

FF

FF

TT

FF

TT

FF

TT

((pp qq) ) ~~rr

TT

FF

TT

TT

TT

TT

TT

pp qq

TT

FF

FF

TT

TT

FF

FF

~r~r

FF

TT

TT

FF

FF

TT

TT

rr

TTFF

TTTT

FFTT

TTFF

FFFF

FFTT

TTTT

qqpp


Construct a truth table for the following compound statement.a.

FFFFFFFFFFFF

FF

FF

TT

FF

TT

FF

TT

FF

FF

TT

FF

FF

FF

TT

FF

FF

FF

FF

TT

FF

TT

TTFFFF

FF

TT

FF

FF

TT

TT

rr

TTTT

FFTT

TTFF

TTFF

FFTT

TTTT

qqpp

Answer:


b. Answer:

FFFFFFFFFFFF

TT

FF

TT

FF

TT

TT

TT

TT

TT

TT

FF

TT

TT

TT

TT

FF

TT

TT

TT

TT

TT

TTFFFF

FF

TT

FF

FF

TT

TT

rr

TTTT

FFTT

TTFF

TTFF

FFTT

TTTT

qqpp

Construct a truth table for the following compound statement.


c. Answer:

FFFFFFFFFFFF

TT

FF

TT

TT

TT

TT

TT

FF

FF

TT

FF

FF

FF

TT

TT

FF

TT

TT

TT

TT

TT

TTFFFF

FF

TT

FF

FF

TT

TT

rr

TTTT

FFTT

TTFF

TTFF

FFTT

TTTT

qqpp

Construct a truth table for the following compound statement.


AssignmentAssignment

Geometry:Geometry:

Pg. 72 – 73 Pg. 72 – 73

#18 – 32, 34, 36, 38, 41 – 44#18 – 32, 34, 36, 38, 41 – 44

Pre-AP Geometry:Pre-AP Geometry:Pg. 72 – 73Pg. 72 – 73

#18 – 32, 34, 36, 38, 40,#18 – 32, 34, 36, 38, 40,41 – 44, 51 and 5241 – 44, 51 and 52

welcome to interactive chalkboard 2.2 logic. objectives determine truth values of conjunctions and...

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