Logic
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Statement-any sentence that is either true
or false
Truth value-the truth or falsity of a
statement
Negation-has opposite meaning as well as
opposite truth value
Compound Statement-two or more
statements joined
Logic
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
Conjunction (^) – compound statement
formed by combing two or more
statements with the word and
Disjunction (v) - compound statement
formed by combing two or more
statements with the word or
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.
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.
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.
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.
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
Answer: June is the sixth month of the year, and a
turtle is a bird; false.
b.
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.
c.
Answer: A square does not have five sides, and June is
the sixth month of the year; true.
d.
Answer: A turtle is not a bird, and a square has five
sides; 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.
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: 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 each disjunction. Then find its truth value.
p: 6 is an even number.
q: A cow has 12 legs
r: A triangle has 3 sides.
a. p or r
Answer: 6 is an even number, or a triangle as 3 sides;
true.
b.
Answer: A cow does not have 12 legs, or a triangle
does not have 3 sides; true.
DANCING The Venn diagram shows the number of
students enrolled in Monique’s Dance School for
tap, jazz, and ballet classes.
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 + 9 – 9 or 25 students enrolled in
jazz and ballet and not tap.
PETS The Venn diagram shows the number of
students at Manhattan School that have dogs, cats,
and birds as household pets.
a. How many students in
Manhattan School have one
of three types of pets?
Answer: 311
b. How many students have
dogs or cats?
Answer: 280
c. How many students have
dogs, cats, and birds as pets?
Answer: 10
Construct a truth table for
.
Step 1 Make columns with the headings
p, q, ~p, and ~p
p
q
~p
~p
Construct a truth table for
.
Step 2 List the possible combinations of truth values for
p and q.
p
q
T
T
F
F
T
F
T
F
~p
~p
Construct a truth table for
.
Step 3 Use the truth values of p to determine the truth
values of ~p.
p
q
T
T
F
F
T
F
T
F
~p
F
F
T
T
~p
Construct a truth table for
.
Step 4 Use the truth values for ~p and q to write the
truth values for ~p  q.
Answer:
p
q
T
T
F
F
T
F
T
F
~p
F
F
T
T
~p
T
F
T
T
Construct a truth table for
.
Step 1 Make columns with the headings
p, q, r, ~q, ~q  r, and p  (~q  r).
p
q
r
~q
~q  r
p  (~q  r)
Construct a truth table for
.
Step 2 List the possible combinations of truth values for
p, q, and r.
p
q
r
T
T
T
T
F
T
T
T
F
T
F
F
F
T
T
F
F
T
F
T
F
F
F
F
~q
~q  r
p  (~q  r)
Construct a truth table for
.
Step 3 Use the truth values of q to determine the truth
values of ~q.
p
q
r
~q
T
T
T
F
T
F
T
T
T
T
F
F
T
F
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
F
F
T
~q  r
p  (~q  r)
Construct a truth table for
.
Step 4 Use the truth values for ~q and r to write the
truth values for ~q  r.
p
q
r
~q
~q  r
T
T
T
F
F
T
F
T
T
T
T
T
F
F
F
T
F
F
T
F
F
T
T
F
F
F
F
T
T
T
F
T
F
F
F
F
F
F
T
F
p  (~q  r)
Construct a truth table for
.
Step 5 Use the truth values for p and ~q  r to write the
truth values for p  (~q  r).
Answer:
p
q
r
~q
~q  r
p  (~q  r)
T
T
T
F
F
T
T
F
T
T
T
T
T
T
F
F
F
T
T
F
F
T
F
T
F
T
T
F
F
F
F
F
T
T
T
T
F
T
F
F
F
F
F
F
F
T
F
F
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.
p
q
r
~r
pq
(p  q)  ~r
Construct a truth table for (p  q)  ~r.
Step 2 List the possible combinations of truth values for
p, q, and r.
p
q
r
T
T
T
T
F
T
T
T
F
T
F
F
F
T
T
F
F
T
F
T
F
F
F
F
~r
pq
(p  q)  ~r
Construct a truth table for (p  q)  ~r.
Step 3 Use the truth values of r to determine the truth
values of ~r.
p
q
r
~r
T
T
T
F
T
F
T
F
T
T
F
T
T
F
F
T
F
T
T
F
F
F
T
F
F
T
F
T
F
F
F
T
pq
(p  q)  ~r
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.
p
q
r
~r
pq
T
T
T
F
T
T
F
T
F
T
T
T
F
T
T
T
F
F
T
T
F
T
T
F
T
F
F
T
F
F
F
T
F
T
T
F
F
F
T
F
(p  q)  ~r
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:
p
q
r
~r
pq
(p  q)  ~r
T
T
T
F
T
T
T
F
T
F
T
F
T
T
F
T
T
T
T
F
F
T
T
F
F
T
T
F
T
T
F
F
T
F
F
F
F
T
F
T
T
F
F
F
F
T
F
F
Construct a truth table for the following compound
statement.
a.
Answer:
p
q
r
T
T
T
T
T
T
T
F
T
F
F
F
T
T
F
T
F
T
T
F
F
F
F
F
F
T
T
F
T
T
F
F
T
F
F
F
F
T
F
F
F
F
F
F
F
F
F
F
Construct a truth table for the following compound
statement.
b.
Answer:
p
q
r
T
T
T
T
T
T
T
F
T
T
T
T
T
T
F
T
T
T
T
F
F
T
F
F
F
T
T
T
T
T
F
F
T
F
T
F
F
T
F
T
T
T
F
F
F
F
F
F
Construct a truth table for the following compound
statement.
c.
Answer:
p
q
r
T
T
T
T
T
T
T
F
T
T
F
T
T
T
F
T
F
T
T
F
F
T
F
T
F
T
T
T
T
T
F
F
T
F
F
F
F
T
F
T
F
T
F
F
F
F
F
F