|9
3.1 Conjunction
Example 1 :
Given :
Problem :
p : Ann is on the softball team.
q : Paul is on the football team.
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 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.
Now that we have defined a conjunction , we can apply it to
Example 1. The conjunction p∧q is true when both " Ann is on the softball
team " and " Paul is on the football team " are true statements ; otherwise it
is false. 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.
| 10
the truth table for conjunction p∧
∧q
p
q
T
T
F
F
T
F
T
F
p∧q
T
F
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 t he 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 : Construct a truth table for the conjunction "a and b."
Solution :
a
b
T
T
F
F
T
F
T
F
a∧b
| 11
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 true , s is true. 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.
| 12
Example 4 :
Given:
p: The number 11 is
q: The number 17 is
r: The number 23 is
Problem: For each conjunction
it is true or false.
prime.
true
composite.
false
prime.
true
below , write a sentence and indicate if
Solution :
Symbol
1.
p∧q
2.
p∧r
3.
q∧r
Statement
Truth value
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 below:
1. x and y
2. ~x and y
3. ~y and x
| 13
Solution :
x
y
T
T
F
F
T
F
T
F
x∧y
x
y
T
T
F
F
T
F
T
F
~x
~x∧y
x
y
T
T
F
F
T
F
T
F
~y
~y∧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.
3.2 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.
| 14
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 false if and only if both
statements are false ; otherwise it is true .
The truth table for disjunction p∨q
p
q
T
T
F
F
T
F
T
F
p∨q
T
T
T
F
Example 2 : Construct a truth table for the disjunction " a or b."
Solution:
a
b
T
T
F
F
T
F
T
F
a∨b
| 15
Example 3 :
Given :
r : x is divisible by 2.
s : x is divisible by 3.
Problem : What are the truth values of r∨s ?
Solution :
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 true , 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.
| 16
Solution :
Symbol
1.
p q
2.
p r
3.
q r
Statement
Truth value
Example 5 : Complete a truth table for each disjunction below.
1. a or b
2. a or not b
3. not a or b
Solution :
a
b
T
T
F
F
T
F
T
F
a∨b
a
b
T
T
F
F
T
F
T
F
~b
a∨~b
a
b
T
T
F
F
T
F
T
F
~a ~a∨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.
| 17
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
T
T
F
F
T
F
T
F
x∧y
x∨y
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.
| 18
3.3 Conditional Statements
Example 1 :
Given:
Problem:
p : I do my homework.
q : I get my allowance.
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. "
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.
the truth table for conditional p→
→q
p
T
T
F
F
q
T
F
T
F
p
q
T
F
T
T
| 19
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.
| 20
Solution : The conditional a b represents " If the sun is made of gas ,
then 3 is a prime number."
a
b
T
T
F
F
T
F
T
F
a→b
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 , "If Gisele has a math assignment,
then David owns a car.
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.
| 21
Example 4 :
Given :
Problem :
r : 8 is an odd number.
s : 9 is composite.
What is the truth value of r
false
true
s?
Solution: Since hypothesis r is false and conclusion s is true , the
conditional r s is ………………………….
Example 5:
Given :
r : 8 is an odd number.
s : 9 is composite.
Problem : What is the truth value of s
false
true
r?
Solution: Since hypothesis s is true and conclusion r is false , the
conditional s r is ………………………...
Example 6:
Given:
p : 72 is equal to 49.
q : A rectangle does not have 4 sides.
r : The number 6 is even.
s : A square is not a quadrilateral.
Problem: Write each conditional below as a sentence. Then indicate its
truth value.
| 22
Symbol
1.
p→q
2.
q→r
3.
p→r
4.
q→s
Statement
Truth value
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.
Summary :
♥ A conditional statement , symbolized by p q , is an if – then
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.
| 23
3.4 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→q)∧(q→p)
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
T
T
F
F
T
F
T
F
p→q
q→p
(p→q)∧(q→p)
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.
| 24
p
q
T
T
F
F
T
F
T
F
p↔q
T
F
F
T
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 q represent?
Solution : The statement p ↔ q represents the sentence, "A polygon is a triangle
if and only if it has exactly 3 sides."
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 one-way 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.
| 25
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
b.
Solution:
The biconditonal a ↔ b represents the sentence: "x + 2 = 7 if and only if
x = 5." 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
T
T
F
F
T
F
T
F
a↔b
Example 3:
Given :
x : I am breathing
y : I am alive
Problem : Write x ↔ y as a sentence.
Solution : x ↔ y represents the sentence,
| 26
Example 4 :
Given :
r : You passed the exam.
s : You scored 65% or higher.
Problem : Write r ↔ s as a sentence.
Solution : r ↔ s represents the sentence ,
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.
I am breathing if and only if I am alive.
x + 2 = 7 if and only if x = 5.
You passed the exam if and only if 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.
| 27
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 false by the definition of a conditional. The statement
q → p is also false by the same definition. Therefore, the sentence " x + 7 = 11 iff
x = 5 " is not biconditional.
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:
Yes. The statement r→s is true by definition of a conditional. The statement
s→r is also true. Therefore, the sentence "A triangle is isosceles if and only if it has
two congruent (equal) sides" is biconditional.
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.
| 28
Exercise 3
1. If b represents “ My babysister has fangs ” and a represents “ I am afraid ” ,
Write each of the following sentences in symbolic form
1.1 My babysister has fangs and I am not afraid.
1.2 My babysister does not have fangs and I am afraid.
1.3 My babysister has fangs and I am afraid.
1.4 My babysister has fangs and I am not afraid.
2. If p represents “ Jennifer is moving out ” , q represents“ Jason is moving in”
and r represents “ Leena is running away ” , Write each of the following
sentences in symbolic form
2.1 Jennifer is moving out or Leena is running away.
2.2 Jennifer is moving out and Jason is moving in.
| 29
2.3 Either Jennifer is moving out or Jason is moving in and Leena is running
away.
2.4 Jason is not moving in or Jennifer is moving out.
3. If n represents “ I am pursued by a lion ” m represents“ Monkey laugh at me ”
and p represents “ I take shelter in a tree ” , Write each of the following
sentences in symbolic form
3.1 If I am pursued by a lion , then I take shelter in a tree.
3.2 I take shelter in a tree , then Monkey do not laugh at me.
3.3 I take shelter in a tree if Monkey laugh at me or I am pursued by a lion.
3.4 If I am pursued by a lion and do not take shelter in a tree , then Monkey
laugh at me.
| 30
4. If p represents “ Lora is happy ” q represents“ Molly is happy ”
and r represents “ Business is good , Write each of the following sentences in
symbolic form
4.1 Lora is happy if and only if Business is good.
4.2 Business is good if and only if Lora is happy and Molly is happy.
4.3 Lora is happy if Molly is happy , and Molly is happy if Lora is happy.
4.4 Business is not good and Molly is happy if and only if Lora is happy.
5. If p represents “ The number 2 is prime. ” q represents“ The number 2 is even ”
and r represents “ The number 3 is odd , Write each of the following symbolic
form in statement form.
5.1 p ∨ q
5.2 q ∧ ∼r
| 31
5.3 p → q
5.4 ( p ∧ q )→ r
5.5 q ↔ r
Summary:
p
q
T
T
T
F
F
T
F
F
p∧q
p∨q p→q p↔q
∼p
∼q
An order of priority in statement connection is very significant and must be
considered as ( ) , ∼ , ∧ , ∨ , → , ↔ respectively.