| 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 pq
represents the sentence, "Ann is on the softball team and Paul is on the football
team." The statement pq
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 pq. 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 pq 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 pq, we start by listing every combination of truth values in the first two
columns of the truth table below.
| p |
q |
pq |
| T |
T |
|
| T |
F |
|
| F |
T |
|
| F |
F |
|
Next, we complete the last column according to the rules for conjunction listed above.
| p |
q |
pq |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
F |
F |
The truth table above lists the truth values of pq.
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 |
ab |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
F |
F |
|
| Example 3: |
|
 |
|
| Given: |
r: The number x is odd. |
| s: The number x is prime. |
| Problem: |
Can we list all truth values for rs
in a truth table? Why or why not? |
|
| Solution: |
Since each statement given in this example represents an
open sentence,
the truth value of rs
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 rs
in a truth table. We can, however, find the truth value of rs
for given values of x as shown below. |
|
| If x = 3, then r is true, s is true. The conjunction rs is true. |
| If x = 9, then r is true, s is false. The conjunction rs is false. |
| If x = 2, then r is false, s is true. The conjunction rs is false. |
| If x = 6, then r is false, s is false. The conjunction rs is false. |
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. |
|
|
| 1. |
pq |
The number 11 is prime and the number 17 is composite. |
false |
| 2. |
pr |
The number 11 is prime and the number 23 is prime. |
true |
| 3. |
qr |
The number 17 is composite and the number 23 is prime. |
false |
|
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 |
|
|
| Solution: |
| x |
y |
xy |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
F |
F |
|
|
| x |
y |
~x |
~xy |
| T |
T |
F |
F |
| T |
F |
F |
F |
| F |
T |
T |
T |
| F |
F |
T |
F |
|
|
| x |
y |
~y |
~yx |
| T |
T |
F |
F |
| T |
F |
T |
T |
| F |
T |
F |
F |
| F |
F |
T |
F |
|
|
| Summary: |
A conjunction is a compound statement formed by joining two
statements with the connector "and." The conjunction "p and q"
is symbolized by pq.
A conjunction is true when both of its combined parts are true; otherwise it is false. |
 |
Exercises
Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer
is provided in the RESULTS BOX. If you make a mistake, choose a different button.
| 1. | Which of the following
sentences is a conjunction? |
|
|
| 2. | Which of the following
statements is a conjunction? |
|
|
| 3. | A
conjunction is used with which connector? |
|
|
| 4. |
If a is false and b is true, what is the truth value of ab?
|
|
|
| 5. |
| Given: |
r: y is prime. |
| s: y is even. |
| Problem: |
What is the truth value of rs
when y is replaced by 2? |
|
|
|
