Chapter 5: Section 5-2
Truth Tables and Equivalent Statements
D. S. Malik
Creighton University, Omaha, NE
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
1 / 25
(p ^ q ) _ q
The truth table of this statement is:
Consider the statement:
p
T
T
F
F
q
T
F
T
F
q
F
T
F
T
p^ q
F
T
F
F
(p ^
T
F
T
T
q)
(p ^
T
F
T
F
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
q) _ q
2 / 25
Remark
(Number of rows in a truth table) The number of rows in the truth
table of a compound statement depends on the number of variables in the
compound statement. As we have seen, if the number of variables in a
compound statement is 2, then the variables has four choices, so the
number of rows is 22 = 4. Note that these are the rows below the …rst row
that contains column headings.
In general, if a compound statement has n variables, then the number of
rows, below the columns headings row, is 2n.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
3 / 25
Example
Let p, q, and r be statements. We construct the truth table for the
compound statement (s p ^ q ) ! r .
Because the compound statement (s p ^ q ) ! r has 3 variables, the
number of rows in the truth table is 23 = 8. The truth table for
(s p ^ q ) ! r is
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
sp
F
F
F
F
T
T
T
T
s p^q
F
F
F
F
T
T
F
F
(s p ^ q ) ! r
T
T
T
T
T
F
T
T
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
4 / 25
Example
Let A be the statement (s p ^ q ) ! (s (q ! p )). We construct the
truth table for A. This statement contains two statement variables. So, to
construct the truth table for A we have to consider four di¤erent
assignments of truth values. The following is the truth table of A.
p
T
T
F
F
q
T
F
F
T
sp
F
F
T
T
(s p ^ q )
F
F
F
T
q!p
T
T
T
F
s (q ! p )
F
F
F
T
A
T
T
T
T
From the truth table it follows that the truth value of A is T for any
assignments of truth values T and F to p and q. Such a statement is
called a tautology.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
5 / 25
De…nition
(i) A statement A is said to be a tautology if the truth value of A is T for
any assignment of the truth values T and F to the statement(s) occurring
in A.
(ii) A statement A is said to be a contradiction if the truth value of A is
F for any assignment of the truth values T and F to the statement(s)
occurring in A.
Notation: For a statement A, we use the notation
A is a tautology.
A to indicate that
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
6 / 25
Logically Equivalent Statements
The truth table of the statement (p ^ (p ! q )) ! q is:
p
T
T
F
F
q
T
F
T
F
p!q
T
F
T
T
p ^ (p ! q )
T
F
F
F
(p ^ (p ! q )) ! q
T
T
T
T
From the table it follows that (p ^ (p ! q )) ! q is a tautology.
In this case, we say that the statement (p ^ (p ! q )) logically
implies the statement q.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
7 / 25
Example
Consider the statements (p _ q ) and s p ^ s q. The truth table of
A = (p _ q ) !s p ^ s q is
p
T
T
F
F
q
T
F
T
F
This implies that
p_q
T
T
T
F
s (p _ q )
F
F
F
T
sp
F
F
T
T
sq
F
T
F
T
s p^ s q
F
F
F
T
A
T
T
T
T
(p _ q ) logically implies s p ^ s q.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
8 / 25
Example
The truth table of B = (s p ^ s q ) !
p
T
T
F
F
q
T
F
T
F
sp
F
F
T
T
sq
F
T
F
T
s p^ s q
F
F
F
T
(p _ q ) is
p_q
T
T
T
F
This implies that s p ^ s q logically implies
s (p _ q )
F
F
F
T
B
T
T
T
T
(p _ q ).
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
9 / 25
De…nition
A statement A is said to be logically equivalent to a statement B if the
statement A logically implies B and the statement B logically implies A,
i.e., A ! B and B ! A are tautologies. If A is logically equivalent to B,
then symbolically we write A B.
Remark
Let A and B be (compound) statements. To show that A B, we can
show that A ! B and B ! A are tautologies or equivalently we can
construct the truth tables of the statements A and B and show that the
columns labeled A and B are the same. Note that when you use truth
tables to show that two statements are equivalent, you can construct the
truth tables of both the statements in the same table. However, when you
separately construct the truth tables for A and B, then the columns
labeled by the variables in the statements A and B must list the truth
values of the variables in the same order.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
10 / 25
Example
In this example, we show that the implication p ! q is equivalent to
s p _ q. For this we construct the truth table for (p ! q ) and s p _ q.
p
T
T
F
F
q
T
F
T
F
sp
F
F
T
T
p!q
T
F
T
T
s p_q
T
F
T
T
From the last two columns of this table, it follows that (p ! q ) and are
s p _ q are equivalent.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
11 / 25
Example
Consider the statement: ((p ! q )^
statement is:
p
T
T
F
F
q
T
F
T
F
p!q
T
F
T
T
q
F
T
F
T
(p ! q )^
F
F
F
T
This implies that ((p ! q )^ q ) !
(p ! q )^ q logically implies p
q) !
q
p. The truth table of this
p
F
F
T
T
((p ! q )^ q ) !
T
T
T
T
p
p is a tautology. Hence,
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
12 / 25
De Morgan’s Laws and the Negation of Conjunctions and
Disjunctions
Theorem
( De Morgan’s laws) Let p and q be statements. Then
s (p ^ q )
s (p _ q )
(s p ) _ (s q ),
(s p ) ^ (s q ).
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
13 / 25
Example
Let p : “Today is Sunday” ; q : “It is a nice day to go for a walk.”
Then, in words, the statement p ^ q is
Today is Sunday and it is a nice day to go for a walk.
Now
s (p ^ q )
(s p ) _ (s q ),
s p : Today is not Sunday,
s q : It is not a nice day to go for a walk.
(s p ) _ (s q ) : Today is not Sunday or it is not a nice day to go for a walk.
Thus, the negation of the statement “Today is Sunday and it is a nice day
to go for a walk” is the statement
Today is not Sunday or it is not a nice day to go for a walk.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
14 / 25
Example
Consider the statement
Amanda will not win the marathon or her husband will not buy her a car.
We write the negation of this statement.
The given statement consists of the statements “Amanda will not win the
marathon” and “Her husband will not buy her a car.” These statements
are connected by using the connective or. So we write the negation of
each of these statements and then connect them using the connective and.
Hence, the negation of the given compound statement is
Amanda will win the marathon and her husband will buy her a car.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
15 / 25
Negation of an Implication (Conditional Statement)
Consider the compound statement
“If I get a penny every time you lie, then I will be very rich.”
Suppose that we want to negate this statement. Let p : “I get a penny
every time you lie”; q : “I will be very rich”. Then in symbol form the
given statement is
p ! q.
We want to determine
p!q
This implies that
By the DeMorgan’s law
(
Hence,
(p ! q ). Recall that
(
(p ! q )
p _ q) =
(
p _ q)
(
p) ^ (
(p ! q )
p^
p _ q)
q) = p^
q.
q
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
16 / 25
(p ! q )
p^
q
Now p is the statement “I get a penny every time you lie” and q is the
statement “I will not be very rich”. Hence, the negation of the statement
“If I get a penny every time you lie, then I will be very rich” is the
statement
I get a penny every time you lie and I will not be very rich.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
17 / 25
To write the negation of the implication, p ! q, we do the following:
1
Negate the conclusion (consequent).
2
Connect the hypothesis (antecedent), p, and the negated conclusion
(consequent), q, using the logical connective ‘and’.
That is:
(p ! q )
p^
q
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
18 / 25
Example
Consider the conditional statement
If I pay my taxes on time, then I will not be penalized.
We write the negation of this statement.
Here the hypothesis is “I pay my taxes on time” and the conclusion is “I
will not be penalized.” The negation of the conclusion is “I will be
penalized.” Hence, the negation of the given statement is
I pay my taxes on time and I will be penalized.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
19 / 25
More on Implications
Consider the statement: “Orange juice contains vitamin C .”
This statement is an implication and can be written as
“If it is orange juice, then it contains vitamin C .”
This means that there is more than one way to write an implication.
Let us consider some more examples.
Example
(i) Consider the statement: “July 4th is a federal holiday.” Using if...then
form, this statement can be written as:
If it is July 4th, then it is a federal holiday.
(ii) Consider the statement: Every doctor carries a stethoscope.The
equivalent statement is:
If he/she is a doctor, then he/she carries a stethoscope.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
20 / 25
De…nition
Let p and q be statements.
(i) The statement q ! p is called the converse of the implication p ! q.
(ii) The statement s p ! s q is called the inverse of the implication
p ! q.
(iii) The statement s q ! s p is called the contrapositive of the
implication p ! q.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
21 / 25
Example
Consider the statement “If today is Sunday, then I will go for a walk.” Let
p and q be the following statements:
p : Today is Sunday.
q : I will go for a walk
Then the given statement can be written as p ! q. The converse of this
implication is q ! p, which is
q ! p : If I will go for a walk, then today is Sunday.
The inverse of the above implication is s p ! s q, which is
s p !s q : If today is not Sunday, then I will not go for a walk.
The contrapositive of the above implication is s q ! s p, which is
s q !s p : If I will not go for a walk, then today is not Sunday.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
22 / 25
Exercise: Determine whether the statements s p ^ s q and
s (s p ! q ) are equivalent.
Solution: Now
p
T
T
F
F
q
T
F
T
F
p
F
F
T
T
sq
F
T
F
T
s p^ s q
F
F
F
T
sp!q
T
T
T
F
s (s p ! q )
F
F
F
T
From the truth table, it follows that the statements
s p ^ s q and s (s p ! q ) are equivalent
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
23 / 25
Exercise: Use De Morgan’s law to write the negation of the statement:
This is New Year’s Day and it is very cold.
Solution: The statement “This is New Year’s Day and it is very cold”
contains the statements “This is New Year’s Day” and “It is
very cold,” and the logical connective ‘and’.
The negation of the statement “This is New Year’s Day” is
the statement “This is not New Year’s Day.” Also the
negation of the statement “It is very cold” is the statement
“It is not very cold.” Now, because (p ^ q )
p _ q,
it follows that the negation of the given statement is
This is not New Year’s Day or it is not very cold.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
24 / 25
Exercise: Write the inverse, converse, and the contrapositive of the
following statement:
If it is a rose, then it is a ‡ower.
Solution: Let p : It is a rose; and q : It is a ‡ower.
In symbols, the statement “If it is a rose, then it is a ‡ower”
is:
p ! q.
The converse of this implication is q ! p, which is
q ! p : If it is a ‡ower, then it is a rose.
The inverse of the above implication is s p ! s q, which is
s p !s q : If it is not a rose, then it is not a ‡ower.
The contrapositive of the above implication is s q ! s p,
which is
s q !s p : If it is not a ‡ower, then it is not a rose.
D. S. Malik Creighton University, Omaha, NEChapter
()
5: Section 5-2 Truth Tables and Equivalent Statements
25 / 25