Name _____________________________
Logic Unit Topics
UQ: How do you use truth tables to determine the validity of a statement?
 Negation
EQ: How do you negate a statement?
 Conjunction
EQ: What is a conjunction and a disjunction?
 Disjunction
 Conditional
EQ: What is a conditional and its converse, inverse, and contrapositive?
 Compound Statements EQ: How do you determine the truth value of a compound statement?
 Biconditionals
EQ: What is a biconditional and how is it expressed?
 Tautologies
EQ: How do you determine if a statement is a tautology?
 Equivalent Statements EQ: How do you determine if two statements are logically equivalent?
Lesson on Sentences and Negation
Example 1: Examine the sentences below.
1. Every triangle has three sides.
2. Albany is the capital of New York State.
3. No prime number is even.
Each of these sentences is a closed sentence.
Definition: A closed sentence is an objective statement which is either ___________ or ____________.
Thus, each closed sentence in Example 1 has a truth value of either true or false. Indicate it’s
truth value below.
1. Every triangle has three sides.
2. Albany is the capital of New York State.
3. No prime number is even.
Note that the third sentence is false since 2 is a prime number. It is possible that a closed
sentence will have different truth values at different times. This is demonstrated in Example 2
below.
Example 2:
1. Today is Tuesday.
2. Bill Clinton is the President of the United States.
Example 3: Examine the sentences below.
1. x + 3 = 7
2. She passed math.
3. y - 4 = 11
4. He is my brother.
The sentences in Example 3 are open sentences.
Definition: An open sentence is a statement which contains a _______________ and becomes either
true or false depending on the value that replaces the variable.
Let's take another look at Example 3. This time we will identify the variable for each open
sentence.
1. x + 3 = 7
The variable is
2. She passed math. The variable is
3. y - 4 = 11
The variable is
4. He is my brother. The variable is
Now that we have identified the variables, we can analyze the meaning of these open sentences.
Sentence 1 is true if x is replaced by __________, but false if x is replaced by ______________.
Sentence 2 is either true or false depending on the value of the variable "she."
Sentence 3 is true if y is replaced by 15, but false otherwise.
Sentence 4 is either true or false depending on the value of the variable "he."
In summary, the truth value of each open sentence depends on what value is used to replace the
variable in that sentence.
The negation of a statement is indicated by a ~ placed in front of the letter representing the
statement.
Example 4:
Given: Let p represent, "Baseball is a sport."
Let q represent, "There are 100 cents in a dollar."
Let r represent, "She does her homework."
Let s represent, "A dime is not a coin."
Problem: Write each sentence below using symbols and indicate if it is true, false or open.
Symbols T/F/Open
1. A dime is a coin.
2. Baseball is not a sport.
3. She does her homework.
4. There are not 100 cents in a dollar.
5. She does not do her homework.
6. Baseball is a sport.
Example 5:
Given:
Let p represent the closed sentence "The number 9 is odd."
Problem: What does ~p mean?
In Example 5 we are asked to find the negation of p.
Definition: The negation of statement p is "not p." The negation of p is symbolized by "~p." The truth
value of ~p is the opposite of the truth value of p.
Solution:
Statement
T/F
p: The number 9 is odd.
~p:
Let's look at some more examples of negation.
Example 6:
Statement
T/F
r: 7 < 5
~r:
Example 7:
Statement
T/F
a: The product of two negative numbers is a positive number.
~a:
We can construct a truth table to determine all possible truth values of a statement and its
negation.
Definition:
A truth table helps us find all possible truth values of a
statement. Each statement is either True (T) or False
(F), but not both.
Connection:
To help us remember this definition, think of a
computer, which is either on or off, but not both.
Example 8: Construct a truth table for the negation of x.
Solution:
x ~x
T F
F T
In Example 8, when x is true, ~x is false; and when x is false, ~x is true. From this truth table, we
can see that a statement and its negation have opposite truth values.
Example 9: Construct a truth table for the negation of p.
Solution:
p ~p
T
F
We can also negate a negation. For example, the negation of ~p is ~(~p) or p. This is illustrated
in the example below.
Example 10: Construct a truth table for the negation of p, and for the negation of not p.
Solution:
p ~p
~(~p)
T
F
Summary:
A statement is a sentence that is either true or
false. A closed sentence is an objective statement
which is either true or false. An open sentence is a
statement which contains a variable and becomes
either true or false depending on the value that
replaces the variable. The negation of statement p
is "not p", symbolized by "~p". A statement and its
negation have opposite truth values.
Exercises on Sentences and Negation
Directions: Read each question below and select your answer.
1. Which of the following is a closed
sentence?
Summer follows spring.
A quarter is a coin.
There are 360 days in a year.
All of the above.
2. What is the negation of, "Jenny rides the
bus"?
Jenny does not like to ride the bus.
Jenny does not ride the bus.
Jenny likes to ride the bus.
None of the above.
3. Which of the following is the negation of
x?
-x
~(~x)
~x
None of the above
4. Given:
a: A triangle is not a polygon.
b: A square is a rectangle.
Problem: Which of the following is the
negation of "A triangle is not
a polygon"?
~(~b)
~a
a
None of the above.
5. Which of the following is an open
sentence?
The number 4 is even.
The number 8 is odd.
The number 5 is even.
The number x is odd.
Lesson on Conjunction
Example 1:
Given:
p: Ann is on the softball team.
q: Paul is on the football team.
Problem: What does p
q represent?
Solution:
In Example 1, statement p represents the sentence, "Ann is on the softball team," and statement q
represents the sentence, "Paul is on the football team." The symbol is a logical connector which
means "and." Therefore, the compound statement p q represents the sentence, "Ann is on the
softball team and Paul is on the football team." The statement p q is a conjunction.
Definition: A conjunction is a compound statement formed by joining two statements with the
connector ________. The conjunction "p and q" is symbolized by p q. A conjunction is
true when both of its combined parts are ____________; otherwise it is _____________.
We can construct a truth table for the conjunction "p and q." In order to list all truth values of p
q, we start by listing every combination of truth values in the first two columns of the truth
table below.
p q p
q
T T
T F
F T
F F
The truth table above lists the truth values of p q. A truth table is an excellent tool for listing
the truth values of a conjunction (or any compound statement). (Note: Throughout our lessons on
symbolic logic, we will always construct truth tables with the first two columns listed exactly as
above. The order of the truth values in these first columns is critical to finding all truth values for
a given statement. This order will also apply to other formats used to list truth values in more
advanced lessons.) Let's look at some more examples of conjunction.
Example 2:
Given:
a: A square is a quadrilateral.
b: Harrison Ford is an American actor.
Problem: Construct a truth table for the conjunction "a and b."
Solution:
a
b
a
b
Example 3:
Given:
r: The number x is odd.
s: The number x is prime.
Problem: Can we list all truth values for r
s in a truth table? Why or why not?
Solution: Since each statement given in this example represents an open sentence, the truth value of r
s will depend on the value of variable x. But there are an infinite number of replacement
values for x, so we cannot list all truth values for r s in a truth table. We can, however, find
the truth value of r s for given values of x as shown below.
If x = 3, then r is _________, s is _________. The conjunction r
s is _________.
If x = 9, then r is _________, s is _________. The conjunction r
s is _________.
If x = 2, then r is _________, s is _________. The conjunction r
s is _________.
If x = 6, then r is _________, s is _________. The conjunction r
s is _________.
In the next example we are given the truth values of each statement. We are then asked to
determine the truth values of the specified conjunctions.
Example 4:
Given:
p: The number 11 is prime.
true
q: The number 17 is composite.
false
r: The number 23 is prime.
true
Problem: For each conjunction below, write a sentence and indicate if it is true or false.
Statement
1. p
q
2. p
r
3. q
r
T/F
A conjunction is formed by combining two statements with the connector "and." One of these
statements can be a negation as shown in the example below.
Example 5: Construct a truth table for each conjunction : x and y, ~x and y, ~y and x.
Solution:
x y x
y
x y ~x ~x
y
x y ~y ~y
T T
T T
T T
T F
T F
T F
F T
F T
F T
F F
F F
F F
x
Summary: A conjunction is a compound statement formed by
joining two statements with the connector
"and." The conjunction "p and q" is symbolized by p
q. A conjunction is true when both of its
combined parts are true; otherwise it is false.
Exercises on Conjunction
Directions: Read each question below and select your answer.
1. Which of the following sentences is a
conjunction?
Jill eats pizza or Sam eats pretzels.
Jill eats pizza but not pretzels.
Jill eats pizza and Sam eats pretzels.
None of the above.
2. Which of the following statements is a
conjunction?
p+q
p
q
~p
None of the above.
3. A conjunction is used with which
connector?
Not
Or
And
None of the above.
4. If a is false and b is true, what is the truth value of a
True
False
Not enough information was given.
None of the above.
5. Given:
r: y is prime.
s: y is even.
Problem: What is the truth value of r
s when y is replaced by 2?
True
False
Not enough information was given.
None of the above.
b?
Lesson on Disjunction
Example 1:
Given:
p: Ann is on the softball team.
q: Paul is on the football team.
Problem: What does p
q represent?
Solution:
In Example 1, statement p represents, "Ann is on the softball team" and statement q represents, "Paul
is on the football team." The symbol is a logical connector which means "or." Thus, the compound
statement p q represents the sentence, "Ann is on the softball team or Paul is on the football
team." The statement p q is a disjunction.
Definition: A disjunction is a compound statement formed by joining two statements with the
connector OR. The disjunction "p or q" is symbolized by p q. A disjunction is __________
if and only if both statements are ___________; otherwise it is __________.
Complete the truth values of p
p q p
q in the truth table below.
q
T T
T F
F T
F F
Example 2:
Solution:
Given:
a: A square is a quadrilateral.
a ba
b: Harrison Ford is an American actor.
T T
Problem: Construct a truth table for the disjunction "a or b."
T F
F T
F F
b
Example 3:
Given:
r: x is divisible by 2.
s: x is divisible by 3.
Problem: What are the truth values of r
Solution:
s?
Each statement given in this example represents an open sentence, so the truth
value of r s will depend on the replacement values of x as shown below.
If x = 6, then r is _________, and s is _________. The disjunction r
s is _________.
If x = 8, then r is _________, and s is _________. The disjunction r
s is _________.
If x = 15, then r is _________, and s is _________. The disjunction r
s is _________.
If x = 11, then r is _________, and s is _________. The disjunction r
s is _________.
Example 4:
Given:
p: 12 is prime.
false
q: 17 is prime.
true
r: 19 is composite.
false
Problem: Write a sentence for each disjunction below. Then indicate if it is true or false.
1. p
q
2. p
r
3. q
r
Example 5:
Complete a truth table for each disjunction below.
1. a or b
2. a or not b
3. not a or b
a b a
b
a b ~b a
~b
a b ~a ~a
T T
T T
T T
T F
T F
T F
F T
F T
F T
F F
F F
F F
b
Students sometimes confuse conjunction and disjunction. Let's look at an example in which
we compare the truth values of both of these compound statements.
Example 6:
Given:
x: Jayne played tennis.
y: Chris played softball.
Problem: Construct a truth table for conjunction "x and y" and disjunction
"x or y."
Solution:
x y x
y x
y
T T
T F
F T
F F
With a conjunction, both statements must be true for the conjunction to be true; but with a
disjunction, both statements must be false for the disjunction to be false. One way to
remember this is with the following mnemonic: 'And’ points up to the sand on top of the
beach, while ‘or’ points down to the ore deep in the ground.
Summary: A disjunction is a compound statement formed by joining two statements with the
connector OR. The disjunction "p or q" is symbolized by p q. A disjunction is false if and
only if both statements are false; otherwise it is true.
Exercises on Disjunction
Directions: Read each question below and select your answer.
1. Which of the following sentences is a
disjunction?
Amy played soccer or Bill played
hockey.
Amy played soccer and Bill played
hockey.
Amy did not play soccer and Bill
played hockey.
None of the above.
2. Which of the following statements is a
disjunction?
~x
y
x
y
x
y
None of the above.
3. A disjunction is used with which
connector?
And
Or
Not
None of the above.
4. If a is false and b is true, what is the truth
value of a ~b?
True
False
Not enough information was given
None of the above.
5. Given:
r: y is prime.
s: y is even.
Problem: Which of the following is a true
statement when y is replaced
by 3?
r
~s
r
~s
r
s
All of the above.
Lesson on Conditional Statements
Example 1:
Given:
p: I do my homework.
q: I get my allowance.
Problem: What does p
q represent?
Solution:
In Example 1, p represents, "I do my homework," and q represents "I get my allowance." The
statement p q is a conditional statement which represents "If p, then q" which is …
Definition:
A conditional statement, symbolized by p q, is an if-then statement in which p is a
hypothesis and q is a conclusion. The logical connector in a conditional statement is
denoted by the symbol
. The conditional is defined to be true unless a true hypothesis
leads to a false conclusion. Complete the truth table for p q below.
p q p
q
T T
T F
F T
F F
In the truth table above, p q is only false when the hypothesis (p) is true and the conclusion (q)
is false; otherwise it is true. Note that a conditional is a compound statement. Now that we have
defined a conditional, we can apply it to Example 1.
Example 1:
Given:
p: I do my homework.
q: I get my allowance.
Problem: What does p
q represent?
Solution:
In Example 1, the sentence, "I do my homework" is the hypothesis and the sentence, "I get my
allowance" is the conclusion. Thus, the conditional p q represents the hypothetical proposition, "If I
do my homework, then I get an allowance." However, as you can see from the truth table above,
doing your homework does not guarantee that you will get an allowance! In other words, there is not
always a cause-and-effect relationship between the hypothesis and conclusion of a conditional
statement.
Example 2:
Given:
a: The sun is made of gas.
b: 3 is a prime number.
Problem: Write a
b as a sentence. Then construct a truth table for this conditional.
Solution: The conditional a
a b a
b represents …
b
T T
T F
F T
F F
In Example 2, "The sun is made of gas" is the hypothesis and "3 is a prime number" is the conclusion.
Note that the logical meaning of this conditional statement is not the same as its intuitive meaning. In
logic, the conditional is defined to be true unless a true hypothesis leads to a false conclusion. The
implication of a b is that: since the sun is made of gas, this makes 3 a prime number. However,
intuitively, we know that this is false because the sun and the number three have nothing to do with
one another! Therefore, the logical conditional allows implications to be true even when the
hypothesis and the conclusion have no logical connection.
Example 3:
Given:
x: Gisele has a math assignment.
y: David owns a car.
Problem: Write x
y as a sentence.
Solution: The conditional x
y represents, …
In the following examples, we are given the truth values of the hypothesis and the
conclusion and asked to determine the truth value of the conditional.
Example 4:
Given:
r: 8 is an odd number.
false
s: 9 is composite.
true
Problem: What is the truth value of r
s?
Solution:
Example 5:
Given:
r: 8 is an odd number.
false
s: 9 is composite.
true
Problem: What is the truth value of s
Solution:
r?
Example 6:
Given:
p: 72 = 49.
true
q: A rectangle does not have 4 sides.
false
r: Harrison Ford is an American actor.
true
s: A square is not a quadrilateral.
false
Problem Write each conditional below as a sentence. Then indicate its truth
:
value.
1. p
q
2. q
r
3. p
r
4. q
s
5. r
~p
6. ~r
p
Note that in item 5, the conclusion is the negation of p. Also, in item 6, the hypothesis is the
negation of r.
Example 7:
Every conditional p
The converse of p
The inverse of p
q has 3 related conditional statements.
q is q
q is ~p
The contrapositive of p
p.
~q.
q is ~q
~p.
Complete the truth table to determine how the various conditionals relate.
p q p
q q
p ~p
~q ~q
~p
T T
T F
F T
F F
Note: Which ones have the same truth values and which ones do not?
Summary:
A conditional statement, symbolized by p q, is an ifthen statement in which p is a hypothesis and q is a
conclusion. The conditional is defined to be true unless a
true hypothesis leads to a false conclusion.
Exercises on Conditional Statements
Directions: Read each question below and select your answer.
1. Which of the following is a conditional statement?
Amy plays soccer or Bill plays hockey.
Bill plays hockey when Amy plays soccer.
If Amy plays soccer then Bill plays hockey.
None of the above.
2. Given:
r: You give me twenty dollars.
s: I will be your best friend.
Problem: Which of the following statements
represents, "If you give me twenty
dollars, then I will be your best friend"?
r
s
r
s
s
r
None of the above.
3. What is the truth value of r
s when the
hypothesis is false and the conclusion is true in
Example 2?
True
False
Not enough information was given.
None of the above.
4. Given:
a: x is prime.
b: x is odd.
Problem: What is the truth value of a
= 2?
b when x
True
False
Not enough information was given.
None of the above.
5. What is the truth value of a
Exercise 4?
b when x = 9 in
True
False
Not enough information was given.
None of the above.
Lesson on Compound Statements
Now that we have learned about negation, conjunction, disjunction and the conditional, we can
include the logical connector for each of these statements in more elaborate statements. In this
lesson, we will learn how to determine the truth values of a compound statement with the logical
connectors ~, , and
.
Example 1:
Given:
p: 72 = 49
true
q: A rectangle does not have 4 sides.
false
r: Harrison Ford is an American actor.
true
Problem Write each sentence below in symbolic form. Then determine its truth
:
value.
1. If 72 = 49, then a rectangle has 4 sides.
2. If 72
49, then a rectangle does not have 4 sides.
3. If a rectangle has 4 sides, then Harrison Ford is not an American actor.
4. If Harrison Ford is an American actor, then 72
49.
5. If 72 = 49 or a rectangle does not have 4 sides, then Harrison Ford is not an American
actor.
In Example 1, each of the first four sentences is represented by a conditional statement in
symbolic form. In item 5, (p q) ~r is a compound statement that includes the connectors
, and ~. It is easier to determine the truth value of such an elaborate compound statement
when a truth table is constructed as shown below.
p q ~r p
T F F
q (p
T
q)
F
~r
,
Example 2:
Given:
p: 28 is a multiple of 7.
true
q: 7 is an even number.
false
Problem: Determine the truth value of this compound statement: ~p
(q
p)
c)
b
Solution:
p
q
~p
q
p
~p
(q
p)
Example 3:
Given:
a is true, b is false, and c is true.
Problem: Determine the truth value of this compound statement: (~a
Solution:
a
b
c
~a
~a
c (~a
c)
b
In the examples above, we were given the truth values of each sentence and asked to determine
the truth value of the resulting compound statement. However, when we are not given this
information, we need to construct a truth table. In each of the following examples, we will
construct a truth table for the given compound statement in order to determine its truth values.
Example 4: What are the truth values of this compound statement? (p q)
p q p
T T
T F
F T
F F
q (p
q)
q
q
Example 5: What are the truth values of this compound statement? q
p q ~q
p
~q q
(p
(p
~q)
~q)
T T
T F
F T
F F
Example 6: What are the truth values of this compound statement? (s
r s ~r
s
r
(s
r)
r) ~r
~r
T T
T F
F T
F F
Example 7: What are the truth values of this compound statement? ~b
a b ~b
a
b
~b
(a
b)
T T
T F
F T
F F
Summary:
We have learned how to write a sentence as a compound
statement in symbolic form. We have learned how to determine
the truth values of a compound statement with the logical
connectors ~, , and
.
(a b)
Exercises on Compound Statements
Directions: Read each question below, work out the truth table and select your answer.
1. Given:
a: 11 is prime.
b: 11 is odd.
Problem: Which of the following sentences
represents (a b) ~b?
If 11 is prime and 11 is odd, then 11 is not odd.
If 11 is prime or 11 is not odd, then 11 is not
odd.
If 11 is prime or 11 is odd, then 11 is not odd.
None of the above.
2. If r and s are false statements, then what is the
truth value of (~r s) s?
True
False
Not enough information was given.
None of the above.
3. If x and y are true statements, then what is the
truth value of (x y) ~y?
True
False
Not enough information was given.
None of the above.
4. What are the truth values of this statement? (~x
y) y
{T, T, T, F}
{T, T, T, T}
{T, F, T, T}
None of the above.
5. What are the truth values of this statement? ~p
(p ~q)
{T, F, T, F}
{F, T, F, T}
{T, T, F, F}
None of the above.
Lesson on Biconditional Statements
Example 1: Examine the sentences below.
Given:
p: A polygon is a triangle.
q: A polygon has exactly 3 sides.
Problem: Determine the truth values of this statement: (p
p)
q)
(q
The compound statement (p q) (q p) is a conjunction of two conditional statements. In the
first conditional, p is the hypothesis and q is the conclusion; in the second conditional, q is the
hypothesis and p is the conclusion. Let's look at a truth table for this compound statement.
p q p
q q
p (p
q)
(q
p)
T T
T F
F T
F F
In the truth table above, when p and q have the same truth values, the compound statement (p
q) (q p) is true. When we combine two conditional statements this way, we have a
biconditional.
Definition: A biconditional statement is defined to be true whenever both parts have the same truth
value. The biconditional operator is denoted by a double-headed arrow
. The
biconditional p
q represents "p if and only if q," where p is a hypothesis and q is a
conclusion. The following is a truth table for biconditional p
q.
p q p
T T
T F
F T
F F
q
In the truth table above, p
q is true when p and q have the same truth values, (i.e., when either
both are true or both are false.) Now that the biconditional has been defined, we can look at a
modified version of Example 1.
Example 1:
Given:
p: A polygon is a triangle.
q: A polygon has exactly 3 sides.
Problem: What does the statement p
Solution: The statement p
q represent?
q represents the sentence, "
Note that in the biconditional above, the hypothesis is: "A polygon is a triangle" and the
conclusion is: "It has exactly 3 sides." It is helpful to think of the biconditional as a conditional
statement that is true in both directions.
Remember that a conditional statement has a oneway arrow ( ) and a biconditional statement has
a two-way arrow (
). We can use an image of a
one-way street to help us remember the symbolic
form of a conditional statement and an image of a
two-way street to help us remember the symbolic
form of a biconditional statement.
Let's look at more examples of the biconditional.
Example 2:
Given:
a: x + 2 = 7
b: x = 5
Problem: Write a
b as a sentence. Then determine its truth values a
Solution:
The biconditonal a
b represents the sentence: "
b.
When x = 5, both a and b are true. When x 5, both a and b are false. A biconditional statement
is defined to be true whenever both parts have the same truth value. Accordingly, the truth values
of a
b are listed in the table below.
a b a
b
T T
T F
F T
F F
So,
Example 3:
Given:
x: I am breathing
y: I am alive
Problem: Write x
Solution: x
y as a sentence.
y represents the sentence, "
Example 4:
Given:
r: You passed the exam.
s: You scored 65% or higher.
Problem: Write r
Solution: r
s as a sentence.
s represents, "
Mathematicians abbreviate "if and only if" with "iff." In Example 5, we will rewrite each
sentence from Examples 1 through 4 using this abbreviation.
Example 5: Rewrite each of the following sentences using "iff" instead of "if and only if."
if and only if
iff
A polygon is a triangle if and only if it has exactly 3
sides.
A polygon is a triangle iff it has exactly 3 sides.
I am breathing if and only if I am alive.
I am breathing iff I am alive.
x + 2 = 7 if and only if x = 5.
x + 2 = 7 iff x = 5.
You passed the exam if and only if you scored 65% or
higher.
You passed the exam iff you scored 65% or
higher.
When proving the statement p iff q, it is equivalent to proving both of the statements "if p, then
q" and "if q, then p." (In fact, this is exactly what we did in Example 1.) In each of the
following examples, we will determine whether or not the given statement is biconditional using
this method.
Example 6:
Given:
p: x + 7 = 11
q: x = 5
Problem: Is this sentence biconditional? "x + 7 = 11 iff x = 5."
Solution:
Let p
q represent "
Let q
p represent "
The statement p
q is …
Example 7:
Given:
r: A triangle is isosceles.
s: A triangle has two congruent (equal) sides.
Problem: Is this statement biconditional? "A triangle is isosceles if and only
if it has two congruent (equal) sides."
Solution:
Summary: A biconditional statement is defined to be true whenever both parts have the same truth
value. The biconditional operator is denoted by a double-headed arrow
. The
biconditional p
q represents "p if and only if q," where p is a hypothesis and q is a
conclusion.
Exercises on Biconditionals
Directions: Read each question below, work it out, and select your answer.
1. Given:
a: y - 6 = 9
b: y = 15
Problem: The biconditional a
b represents
which of the following sentences?
If y - 6 = 9, then y = 15.
y - 6 = 9 if and only if y = 15.
If y = 15, then y - 6 = 9.
None of the above.
2. Given:
r: 11 is prime.
s: 11 is odd.
Problem: The biconditional r
s represents
which of the following sentences?
If 11 is prime, then 11 is odd.
If 11 is odd, then 11 is prime.
11 is prime iff 11 is odd.
None of the above.
3. Given:
x
y
y
x
Problem: If both of these statements are true
then which of the following must also
true?
(x
y)
x
y
(y
x)
x iff y
All of the above.
4. Given:
m
n is biconditional
Problem: Which of the following is a true
statement?
m is the hypothesis
m is the conclusion
n is a conditional statement
n is a biconditional statement
5. Which of the following statements is biconditional?
I am sleeping if and only if I am snoring.
Mary will eat pudding today if and only if it is
custard.
It is raining if and only if it is cloudy.
None of the above.
Lesson on Tautologies
Example 1: What do you notice about each sentence below?
1. A number is even or a number is not even.
2. Cheryl passes math or Cheryl does not pass math.
3. It is raining or it is not raining.
4. A triangle is isosceles or a triangle is not isosceles.
Each sentence in Example 1 is the disjunction of a statement and its negation Each of these
sentences can be written in symbolic form as p ~p. Recall that a disjunction is false if and only
if both statements are false; otherwise it is true. By this definition, p ~p is always true, even
when statement p is false or statement ~p is false! This is illustrated in the truth table below:
p ~p p
~p
T F
F T
The compound statement p ~p consists of the individual statements p and ~p. In the truth table
above, p ~p is always true, regardless of the truth value of the individual statements. Therefore,
we conclude that p ~p is a tautology.
Definition: A compound statement, that is always true regardless of the truth value of the individual
statements, is defined to be a tautology.
Let's look at another example of a tautology.
Example 2: Is (p
q)
p q p
p
T T
T F
F T
F F
q (p
q)
p a tautology?
In the examples below, we will determine whether the given statement is a tautology by
creating a truth table.
Example 3: Is x
x y x
y x
(x
(x
y) a tautology?
y)
T T
T F
F T
F F
Example 4: Is ~b
b ~b ~b
b a tautology?
b
T F
F T
Example 5:
p q (p
T T
T F
F T
F F
q) (p
Is (p
q)
q) (p
(p
q)
q) a tautology?
(p
q)
Example 6: Is [(p
q) p]
p q p
p [(p
q (p
q)
p a tautology?
q)
p]
p
T T
T F
F T
F F
Example 7: Is (r
r s r
s s
r (r
s)
s)
(s
(s
r) a tautology?
r)
T T
T F
F T
F F
Summary:
A compound statement that is always true, regardless of the truth
value of the individual statements, is defined to be a tautology. We
can construct a truth table to determine if a compound statement is
a tautology.
Exercises on Tautologies
Directions: Read each question below, work out the truth table and select your answer.
1.
What is the truth value of r
~r?
True
False
Not enough information was given.
None of the above.
2. Is the following statement a tautology? s
~s
Yes
No
Not enough information was given.
None of the above.
3. Is the following statement a tautology? [(p
~p] q
Yes
No
Not enough information was given.
None of the above.
q)
4. Is the following statement a tautology? ~(x
(~x ~y)
y)
Yes
No
Not enough information was given.
None of the above.
5. Is the following statement a tautology? a
Yes
No
Not enough information was given
None of the above
~a
Lesson on Equivalence
Example 1:
Given:
~p
p
q
If I don't study, then I fail.
I study or I fail.
q
Determine the truth values of the given
statements.
Problem:
Solution:
p q ~p ~p
q p
q
T T
T F
F T
F F
In the truth table above, the last two columns have the same exact truth values! Therefore, the
statement ~p q is logically equivalent to the statement p q.
Definition: When two statements have the same exact truth values, they are said to be logically
equivalent.
Example 2: Construct a truth table for each statement below. Then determine which two are
logically equivalent.
1. ~q
p
2. ~(p
3. p
q)
q
p q ~q ~q
p
p q p
q ~(p
q)
pq p
T T
T T
T T
T F
T F
T F
F T
F T
F T
F F
F F
F F
q
The truth tables above show that ~q p is logically equivalent to p q, since these statements
have the same exact truth values. In Example 3, we will place the truth values of these two
equivalent statements side by side in the same truth table. We will then examine the
biconditional of these statements.
Example 3: Construct a truth table for (~q
p q ~q ~q
p p
q (~q
p)
(p
p)
(p
q)
q)
T T F
T F T
F T F
F F T
The biconditional (~q p) ( p q) is a tautology. This is no coincidence: It turns out that
any two equivalent statements will yield a tautology when placed in the biconditional.
Definition: The biconditional of two equivalent statements is a tautology.
In the next example, we will place the two equivalent statements from Example 1 in the
biconditional.
Example 4:
Given:
~p
p
q If I don't study, then I fail.
q
Problem: Is (~p
I study or I fail.
q)
(p
q) a tautology?
Solution:
p q ~p ~p
T T F
T F F
F T T
F F T
q p
q (~p
q)
(p
q)
Summary:
When two statements have the same exact truth values, they are
said to be logically equivalent. The biconditional of two equivalent
statements is a tautology.
Exercises on Equivalence
Directions: Read each question below. Create a truth table to help you answer each question.
Select your answer.
1. What are the truth values of the following
statement?
(p ~q) ~p
{T, T, T, F}
{T, F, T, T}
{F, T, T, T}
None of the above.
2. Which of the following statements is logically
equivalent to the statement given in Exercise 1?
q
~p
p
p
(p
~q)
q
None of the above.
3. Which of the following statements is logically
equivalent to q (p q)?
q
p
~p
p
(p
~q)
q
None of the above.
4. Which of the following statements is logically
equivalent to a (a b)?
a
b
(a
b)
b
(a
b)
b
None of the above.
5. Given:
Statement x is logically equivalent to
statement y.
Problem: Which of the following is true?
x if and only if y
x
y is a tautology
x iff y
All of the above.
Challenge Exercises: Symbolic Logic
Directions: Read each question below. Create a truth table to help you answer each question.
The answer choices provided correspond to the last column of the truth table for a given
problem. Select your answer.
1. What are the truth values for this statement? ~p
q
{T, F, F, F}
{F, T, T, T}
{F, F, T, F}
None of the above.
2. What are the truth values for this statement? p
~q
{T, T, F, T}
{F, F, T, F}
{F, T, F, F}
None of the above.
3. What are the truth values for this statement? ~a
b
{F, T, T, F}
{T, T, T, F}
{T, F, T, T}
None of the above.
m
4. What are the truth values for this statement? a
~b
{F, T, T, T}
{T, F, T, F}
{T, T, T, F}
None of the above.
5. Choose the word or phrase that best completes this
sentence: The statements in problems 3 and 4 are
____________.
Logically equivalent
Biconditional
Tautologies
None of the above.
6. What are the truth values for this statement? ~q
p
{F, T, T, T}
{T, F, T, F}
{T, T, T, F}
None of the above.
7. Which statement below is logically equivalent to
the statement in problem 6?
p
q
p
q
p
q
None of the above.
8. What are the truth values for this statement? ~r
~(r s)
{T, T, T, T}
{F, F, F, F}
{T, T, T, F}
None of the above.
9. What are the truth values for this statement? (~q
p)
(p q)
{F, T, T, T}
{T, T, T, F}
{T, T, T, T}
None of the above.
10. Choose the word or phrase that best completes
this sentence: The statements given in problems
8 and 9 are _______.
Conjunctions
Tautologies
Triconditionals
None of the above.