The typical conditional statement is \begin{align*}P \rightarrow Q\end{align*} . The inverse, converse and contrapositive are each related to the conditional statement but are not always identical. Which type of statement is always equivalent to the original?
Watch This
http://www.youtube.com/watch?v=IHd8jiUF3Lk James Sousa: The Converse, Contrapositive, and Inverse of an If-Then Statement
Guidance
Consider a conditional statement @$\begin{align*} P \rightarrow Q\end{align*}@$ where:
- @$\begin{align*}P = It \ is \ raining\end{align*}@$ .
- @$\begin{align*}Q = The \ driveway \ is \ wet\end{align*}@$ .
Original Conditional: @$\begin{align*}P \rightarrow Q= \end{align*}@$ If it is raining, then the driveway is wet.
@$\begin{align*}P\end{align*}@$ | @$\begin{align*}Q\end{align*}@$ | @$\begin{align*}P \rightarrow Q\end{align*}@$ |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
The inverse of a conditional negates both the hypothesis and the conclusion.
Inverse: @$\begin{align*}\sim P \rightarrow \ \sim Q= \end{align*}@$ If it is not raining, then the driveway is not wet.
@$\begin{align*}P\end{align*}@$ | @$\begin{align*}Q\end{align*}@$ | @$\begin{align*}\sim P\end{align*}@$ | @$\begin{align*}\sim Q\end{align*}@$ | @$\begin{align*}\sim P \rightarrow \ \sim Q\end{align*}@$ |
T | T | F | F | T |
T | F | F | T | T |
F | T | T | F | F |
F | F | T | T | T |
Notice that the truth values of the inverse are not identical to the truth values of the original conditional. The absence of rain does not guarantee a dry driveway. Some children could have had a water balloon party in the summer. Just because a statement is true does not mean that its inverse will be true!
The converse of a conditional statement switches the order of the hypothesis and the conclusion.
Converse: @$\begin{align*}Q \rightarrow P= \end{align*}@$ If the driveway is wet, then it is raining.
@$\begin{align*}P\end{align*}@$ | @$\begin{align*}Q\end{align*}@$ | @$\begin{align*}Q \rightarrow P\end{align*}@$ |
T | T | T |
T | F | T |
F | T | F |
F | F | T |
Notice that the truth values of the converse are also not identical to the truth values of the original conditional. If children playing with water balloons made the driveway wet then it isn’t necessarily raining. Just because a statement is true does not mean that its converse will be true!
The contrapositive of a conditional statement switches the hypothesis with the conclusion and negates both parts.
Contrapositive: @$\begin{align*}\sim Q \rightarrow \ \sim P= \end{align*}@$ If the driveway is not wet, then it is not raining.
@$\begin{align*}P\end{align*}@$ | @$\begin{align*}Q\end{align*}@$ | @$\begin{align*}\sim P\end{align*}@$ | @$\begin{align*}\sim Q\end{align*}@$ | @$\begin{align*}\sim Q \rightarrow \ \sim P\end{align*}@$ |
T | T | F | F | T |
T | F | F | T | F |
F | T | T | F | T |
F | F | T | T | T |
The contrapositive of a conditional statement is functionally equivalent to the original conditional. This is because you can logically conclude that a dry driveway means no rain. This means that if a statement is a true then its contrapositive will also be true.
Example A
Assume the statement “everyone with blonde hair is smart” is true. Use the contrapositive to write another statement that is related and also true.
Solution: The statement “everyone with blonde hair is smart” can be rewritten as “if a person has blonde hair then the person is smart”. The contrapositive is “if a person is not smart, then the person does not have blonde hair”. This statement must be true if the original statement is true.
Example B
Write the inverse, converse and contrapositive of the following conditional statement.
If you buy our product, then you are attractive.
Solution: Note that advertisers regularly imply certain results about their products that may or may not be true. If you listen carefully you will notice that ironclad conditional statements are always avoided so they are not technically false advertising. At the same time, advertisers prey on the fact that many people mistakenly believe that the inverse and converse are equivalent to the original conditional.
- Inverse: If you do not buy our product, then you are not attractive.
- Converse: If you are attractive, then you will buy our product.
- Contrapositive: If you are not attractive, then you will not buy our product.
Example C
Assume each of the following is true. Do you end up doing your homework?
- If it is raining then you will be tired.
- If you are tired you will nap.
- If you do not do your homework then you will take a nap.
- If you nap or it rains then you will not do your homework.
- If you do not do your homework then it will rain.
- Either it will rain, you will take a nap or you will not be tired.
Solution: Start by identifying the individual statements for each conditional.
- @$\begin{align*}R= It \ is \ raining\end{align*}@$ .
- @$\begin{align*}T= You \ will \ be \ tired\end{align*}@$ .
- @$\begin{align*}H= You \ will \ do \ your \ homework\end{align*}@$ .
- @$\begin{align*}N= You \ will \ take \ a \ nap\end{align*}@$ .
Now, rewrite each sentence using mathematical symbols.
- Statement 1: @$\begin{align*}R \rightarrow T\end{align*}@$
- Statement 2: @$\begin{align*}T \rightarrow N\end{align*}@$
- Statement 3: @$\begin{align*}\sim H \rightarrow N\end{align*}@$
- Statement 4: @$\begin{align*}(N \lor R) \rightarrow \ \sim H\end{align*}@$
- Statement 5: @$\begin{align*}\sim H \rightarrow R\end{align*}@$
- Statement 6: @$\begin{align*}R \lor N \lor T \end{align*}@$
Next, create a train of conditional statements until you reach either “ @$\begin{align*}H\end{align*}@$ ” or “ @$\begin{align*}\sim H\end{align*}@$ ”. From statement 6 you know that either @$\begin{align*}R, N,\end{align*}@$ or @$\begin{align*}T\end{align*}@$ must be true. Consider three cases, one for @$\begin{align*}R\end{align*}@$ being true, one for @$\begin{align*}N\end{align*}@$ being true, and one for @$\begin{align*}T\end{align*}@$ being true.
- Assume @$\begin{align*}R: R \rightarrow N \lor R \rightarrow \ \sim H\end{align*}@$
- Assume @$\begin{align*}N: N \rightarrow N \lor R \rightarrow \ \sim H\end{align*}@$
- Assume @$\begin{align*}T: T \rightarrow N \rightarrow N \lor R \rightarrow \ \sim H\end{align*}@$
In all cases you do not end up doing your homework.
Concept Problem Revisited
The only transformation of a conditional statement that is equivalent to the original statement is the contrapositive. Being comfortable with the contrapositive is absolutely essential for logical reasoning about puzzles and riddles.
Vocabulary
A premise is a starting statement that you use to make logical conclusions.
The inverse of a conditional statement negates both the hypothesis and the conclusion.
The converse of a conditional statement switches the order of the hypothesis and the conclusion.
The contrapositive of a conditional statement switches the hypothesis with the conclusion and negates both parts.
Guided Practice
1. State the inverse, converse, and contrapositive of the following statement.
If a relation passes the vertical line test, then it is a function.
2. You and two logical friends stand in a line so that you cannot see anyone, the friend behind you can see you and the friend in the back can see both people. The three of you are shown three black hats and two white hats. The hats are mixed up and placed on you and your friends’ heads in such a way that nobody knows or can see their own hat. The three of you are told when you know the color of your hat to shout out. Nobody says anything for a long time, but eventually you figure out what color hat you must have even though you cannot see it.
What color hat do you have and how do you know?
3. In a land where some people always lie and some people always tell the truth, you are approached by 2 people who both call the other a liar. You want to know if you are on the right path, so if you can only ask one of them one question what should you ask?
Answers:
1. Inverse: If a relation does not pass the vertical line test, then the relation is not a function.
Converse: If a relation is a function, then the relation passes the vertical line test.
Contrapositive: If a relation is not a function, then the relation does not pass the vertical line test.
2. The way to solve this puzzle is to consider what each person sees starting with the person in the back and then consider what would make them sure and what would make them unsure.
The friend in the back does not shout out. This means that he does not see two white hats because otherwise he would know that he has a black hat.
Fact #1: You and the friend behind you have either BB, BW or WB.
The friend in the middle does not shout out. This person also has figured out Fact #1. This means that if they see a white hat they know they must have a black hat. Since they don’t shout out, they must not see a white hat
Fact #2: You cannot have a white hat.
Conclusion: Your hat is black because that is the only scenario where nobody else is sure about their own hat color.
In this question, both Fact #1 and Fact #2 require contrapositive thinking.
3. This type of question is extremely common in riddles. The trick is to use the fact that a double negative is a positive and a double positive is also a positive. Anything else is negative.
You should ask person A if person B would tell you that you are on the right path.
There will be 4 scenarios that you can organize in a table. Each scenario needs to be carefully thought through. For example, the upper right hand response is yes because if you are on the wrong path the person telling the truth would tell you that no, you are not on the right path. The liar would respond oppositely telling you yes.
Right path | Wrong Path | |
A is liar | No | Yes |
A tells truth | No | Yes |
You can be sure you are on the right path if the response is no and you can be sure you are on the wrong path if the response is yes.
Practice
Assume each statement is true. Use the contrapositive to write another statement that is related and also true.
1. All unripe fruit is bad.
2. All bears like honey.
3. No desserts are healthy.
4. Music by Taylor Swift is good.
5. Everyone who is overweight is unhealthy.
Write the inverse, converse, and contrapositive for each of the following statements.
6. Puppies like to play.
7. If I don’t like something, then I won’t buy it.
8. Everyone at the party is popular.
9. You like music if you go to a concert.
10. All apples have cores.
- My pants are the only things I have that are made of jean material.
- All the clothes you gave me are the right size.
- None of my pants are the right size.
11. Write each of the above statements and its contrapositive symbolically.
12. Determine the final conclusion about “the clothes from you” by stringing the statements/contrapositives together.
- Nobody who is experienced is unsuccessful.
- Mike is always confused.
- No successful person is always confused.
13. Write each of the above statements and its contrapositive symbolically.
14. Determine the final conclusion about “Mike” by stringing the statements/contrapositives together.
- All the plates that got shipped are cracked.
- None of the plates from your mother got shipped.
- Plates that didn’t get shipped should not go in the trash.
15. Write each of the above statements and its contrapositive symbolically.
16. Can you determine whether or not the “plates from your mother” should go in the trash?