Conjunction and Disjunction

As mentioned in the previous file, we can form more complex statements through the use of two simple statements and the words "and" and "or".  The use of those logical connectives are commonly referred to as the conjunction and disjunction of the simple statements.  In order to take into account all the possible truth values for P and Q together, we will need 4 rows to account for all combinations of true and false, as we shall see illustrated below.

Conjunction

For two statements P and Q, the conjunction of P and Q is the statement

P and Q.  

The conjunction "P and Q" is true if both P and Q are true.  This can be illustrated in the following truth table.

P Q P and Q
T T T

T F F
F T F
F F F

Examples of Conjunction of Statements

  1.   Consider the statements

P:  The Eiffel tower is in Budapest.

Q:  The capitol of England is London.

            The conjunction of the two statements P and Q is given by the following

P and Q:  The Eiffel tower is in Budapest and the capitol of England is London.

            In order to check the truth value, notice that P is a false statement while Q is a true statement.  So, we head to the truth table to find our answer.  

P Q P and Q
T T T

T F F
F T F
F F F

            The highlighted row above in the truth table indicates our situation for this example, and we immediately find "P and Q" is a false statement.

      2.   Now take a look at the following statements

P:  5 < 9

Q: 7 is a prime number

           Then the conjunction of P and Q is stated as follows 

P and Q:  5 < 9 and 7 is a prime number

           This gives us two statements P and Q that are both true.  Once again we can refer to the truth table in search of our answer. 

P Q P and Q
T T T

T F F
F T F
F F F

     Notice that the truth table provides with the desired answer again.  As highlighted above, the case where P and Q are true leads to "P and Q" being true.

Disjunction

For two statements P and Q, the disjunction of P and Q is the statement

P or Q.  

The conjunction "P and Q" is true if at least one of P and Q is true.  This can be illustrated in the following truth table.

P Q P or Q
T T T

T F T
F T T
F F F

Examples of Conjunction of Statements

  1.   Consider the statements

P:  The Eiffel tower is in Budapest.

Q:  The capitol of England is London.

            The disjunction of the two statements P and Q is given by the following

P or Q:  The Eiffel tower is in Budapest or the capitol of England is London.

            In order to check the truth value, notice that P is a false statement while Q is a true statement.  So, we head to the truth table to find our answer.  

P Q P or Q
T T T

T F T
F T T
F F F

            The highlighted row above in the truth table indicates our situation for this example, and we immediately find "P or Q" is a true statement.

      2.   Now take a look at the following statements

P:  5 > 9

Q: 7 is a composite number

           Then the disjunction of P and Q is stated as follows 

P or Q:  5 > 9 or 7 is a composite number

           This gives us two statements P and Q that are both true.  Once again we can refer to the truth table in search of our answer. 

P Q P or Q
T T T

T F T
F T T
F F F

     Notice that the truth table provides us with the desired answer again.  As highlighted above, when P and Q are false leads to "P or Q" being false.

Although the truth values may not seem obvious upon first reading the sentences, the truth tables always provide us with the necessary insight to find our desired answers.  So, if we are ever in doubt about a truth value involving a logical connective, it proves useful to consult the corresponding row of a truth table.

Learning Activities

In order to further comprehend the material, please attempt the following exercises.

    1. Find the conjunction of the following statements.  Then find the truth value of the conjunction.
    2. Find the disjunction of the following statements.  Then find the truth value of the disjunction.
    3. Find the truth value of the conjunction of the following statements.
    4. Find the truth value of the conjunction of the following statements.
    5. Find the truth value of the disjunction of the following statements.
    6. Find the truth value of the disjunction of the following statements.