Converse, Inverse, and Contrapositive

With aspects of the implication in our rear view mirror, we now want to form new compound statements from that original implication.  These new statements are called the converse, inverse, and contrapositive statements.

Converse

For two statements P and Q, the converse of the implication "P implies Q" is the statement

Q implies P.  

The converse of "P implies Q" is more commonly written as follows

If Q, then P.

with the truth values of the converse of "P implies Q" given in the last column of the following truth table.

P Q P implies Q Q implies P
T T T

T
T F F T
F T T F
F F T T

After looking at the last two columns of the truth table, we immediately notice that the implication and the converse take on different truth values when there is one simple statement (either P or Q) being true and the other statement being false.  This leads to some confusion at times, however it is important to note the differences between these two compound statements, which we shall explore below.

Examples of the Converse of Implication

  1.   In the previous section we looked at the implication "P implies Q" of the simple statements

P:  Arnold Schwarzenegger created the Olympics.

Q:  The Nile river is located in Africa.

            with our resulting implication being given by

P implies Q:  If Arnold Schwarzenegger created the Olympics, then the Nile river is located in Africa.

            Notice that when creating the converse, we switch the order of the hypothesis and conclusion.  This implies the converse is the following statement

Q implies P:  If the Nile river is located in Africa, then Arnold Schwarzenegger created the Olympics.

     Now we can check the truth value of the converse in our truth table keeping in mind the truth values of Q and P

P Q P implies Q Q implies P
T T T

T
T F F T
F T T F
F F T T

            The highlighted row above in the truth table indicates that the converse of our original implication is a false statement.

      2.    For our second example, let's try to find the converse of the following implication and look for its corresponding truth value

If Washington D.C. is the capitol of the United States, then Barack Obama was the 44th president of the United States.

     Notice that the converse of the above implication would be

If Barack Obama was the 44th president of the United States, then Washington D.C. is the capitol of the United States.

     With our original P and Q statements both being true, we can look to our truth table again to find the truth value of the converse.

P Q P implies Q Q implies P
T T T

T
T F F T
F T T F
F F T T

     Looking in the last column of the above table reveals that the truth value of the converse "Q implies P" is true.

The takeaway from the comparison of the implication and its converse is that there will be places where the truth values agree and, more importantly, there will be places that the truth values differ.  With the implication and its converse being different compound statements, let's search for two compound statements where the truth values will align.      

Inverse

For two statements P and Q, the inverse of the implication "P implies Q" is the statement

(not P) implies (not Q).  

The inverse of "P implies Q" is more commonly written as follows

If not P, then not Q.

with the truth values of the inverse of "P implies Q" given in the last column of the following truth table.

P Q not P not Q (not P) implies (not Q)
T T F

F T
T F F T T
F T T F F
F F T T T

After comparing last truth table to the truth table of the implication (which is included in the last column),

P Q not P not Q (not P) implies (not Q) P implies Q
T T F

F T T
T F F T T F
F T T F F T
F F T T T T

we immediately notice that the implication and the inverse take on different truth values when there is one simple statement (either P or Q) being true and the other statement being false.  This may lead to some disappointment when searching for compound statements that have identical truth values.  However, let us compare the inverse of the implication to the converse of the implication in a single truth table

P Q not P not Q (not P) implies (not Q) Q implies P
T T F

F T T
T F F T T T
F T T F F F
F F T T T T

We have a winner!  The above truth tables show that the converse of an implication will have the exact same truth values as the inverse of the implication regardless of the truth values for P or Q.  This special circumstance is what mathematicians search for and is such a popular concept that is said that the two statements are logically equivalent.  More formally, two compound statements are called logically equivalent if they have the same truth values for all the possible combinations of the simple statements.  The importance of having two logically equivalent statements is that it provides us with two different ways to state the same result.  We shall explore this a bit more in the examples below.

Examples of the Inverse of Implication

  1.   Let's look for the inverse of the following compound statement

If the Empire State building is 1 mile tall, then a triangle has 3 sides.

            Notice that we will need to find the hypothesis and conclusion.  We state both of those simple statements below

P:  The Empire State building is 1 mile tall.

Q:  A triangle has 3 sides.

            In order to move forward from here, we shall find the negation of both P and Q.

not P:  The Empire State building is not 1 mile tall.

not Q:  A triangle does not have 3 sides.

     Putting all of this information together allows us to find the inverse of the original implication as stated below

If the Empire State building is not 1 mile tall, then a triangle does not have 3 sides.

P Q P implies Q not P not Q (not P) implies (not Q)
T T T

F F T
T F F F T T
F T T T F F
F F T T T T

            The highlighted row above in the truth table indicates that the original implication was true, while the inverse of the implication is false.

      2.    For our second example, let's try to find the inverse of the following implication and look for its corresponding truth value

If Paraguay is a country, then Pennsylvania is a continent.

     Notice that the negation of the statements P and Q would lead to the inverse being

If Paraguay is not a country, then Pennsylvania is not a continent.

     Now let's look for the case in our truth table where P was a true statement and Q was false.

P Q not P not Q (not P) implies (not Q)
T T F

F T
T F F T T
F T T F F
F F T T T

     Looking in the last column of the above table reveals that the truth value of the inverse of P implies Q is true.

Although the implication and its inverse may take on different truth values at times, the main takeaway in this section is that the inverse and converse are logically equivalent.  This implies that if we know the truth values of one of those, we will automatically know the truth values for the other.  The final piece to this puzzle is to find a compound statement that takes on the same truth values as the implication.  We shall look no further than the contrapositive of the implication.  

Contrapositive

For two statements P and Q, the contrapositive of the implication "P implies Q" is the statement

(not Q) implies (not P).  

The contrapositive of "P implies Q" is more commonly written as follows

If not Q, then not P.

with the truth values of the contrapositive of "P implies Q" given in the last column of the following truth table.

P Q not Q not P (not Q) implies (not P)
T T F

F T
T F T F F
F T F T T
F F T T T

After looking at the last columns of the truth table, we may recall that the truth values resemble our original implication.  We shall explore that option in the truth table below, which adds on the extra column for the implication

P Q not Q not P (not Q) implies (not P) P implies Q
T T F

F T T
T F T F F F
F T F T T T
F F T T T T

As our intuition led us to believe, the implication and its contrapositive share the same truth values for all combinations of the simple statements P and Q.  As a result, we say that the implication and its contrapositive are logically equivalent.

Examples of the Contrapositive of the Implication

  1.   We shall attempt to find the contrapositive of the following implication 

           If Taylor Swift won a Grammy for album of the year, then Buzz Aldrin landed on the moon.

            In order to create the contrapositive, we will first need to create the negation of both simple statements P and Q

not Q:  Buzz Aldrin did not land on the moon.

not P:  Taylor Swift did not win a Grammy for album of the year

      While it is unnecessary to switch the orders in which we find the negations, we will need to use this switched order when creating the contrapositive

If Buzz Aldrin did not land on the moon, then Taylor Swift did not win a Grammy for album of the year.

      Although this may not seem equivalent to the original implication, we explore our truth table once again to confirm our earlier findings.

P Q not Q not P (not Q) implies (not P) P implies Q
T T F

F T T
T F T F F F
F T F T T T
F F T T T T

            The highlighted row above in the truth table indicates that the contrapositive of our original implication is a true statement.

      2.    For our second example, we shall find the contrapositive of the following statement

If the rupee is currency of the United States , then the United States federal budget is not $500.

     The negation of the two statements below is slightly different than previous situations since the word "not" appears in the Q statement

not Q:  The United States federal budget is $500.

not P:  The rupee is not the currency of the United States.

     Notice that proper sentence structure plays a role as we avoid writing "not not" in the statement "not Q", leading to the above and below statements

If the United States federal budget is $500, then the rupee is not the currency of the United States.

     Rather than look at a truth table this time, we know a false "not Q" statement leads to the contrapositive being a true statement.

As mentioned previously, our focus here is that the implication and its contrapositive will always be logically equivalent, even though the order of the statements may look slightly different.  The importance of this fact is that it provides us with two different methods for stating the same idea.   

Learning Activities

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

    1. Find the converse, inverse, and contrapositive of the implications stated below.  Make sure to label each of your statements accordingly.
    2. Find the truth values for the converse, inverse, and contrapositive of the implications stated below.  Explicitly state which truth value goes to each statement.
    3. Determine which of the following statements are logically equivalent to the implication.
    4. Use a truth table to determine which of the following statements are logically equivalent to "not (A or B)"
    5. Use a truth table to determine which of the following statements are logically equivalent to "not (A and B)