http://www.ck12.org Chapter 16. Logic and Set Theory
Theinverseof a conditional statement negates both the hypothesis and the conclusion.
Theconverseof a conditional statement switches the order of the hypothesis and the conclusion.
Thecontrapositiveof a conditional statement switches the hypothesis with the conclusion and negates both parts.
Guided Practice
- State the inverse, converse, and contrapositive of the following statement.
If a relation passes the vertical line test, then it is a function. - 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? - 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. - 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. - 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.
TABLE16.18:
Right path Wrong Path
A is liar No Yes
A tells truth No Yes