16.4. Inverse, Converse, and Contrapositive http://www.ck12.org
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.
- R=It is raining.
- T=You will be tired.
- H=You will do your homework.
- N=You will take a na p.
Now, rewrite each sentence using mathematical symbols.
- Statement 1:R→T
- Statement 2:T→N
- Statement 3:∼H→N
- Statement 4:(N∨R)→∼H
- Statement 5:∼H→R
- Statement 6:R∨N∨T
Next, create a train of conditional statements until you reach either “H” or “∼H”. From statement 6 you know that
eitherR,N,orTmust be true. Consider three cases, one forRbeing true, one forNbeing true, and one forTbeing
true.
- AssumeR:R→N∨R→∼H
- AssumeN:N→N∨R→∼H
- AssumeT:T→N→N∨R→∼H
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
Apremiseis a starting statement that you use to make logical conclusions.