Many people talk about the benefits of positivity, being more positive in our daily lives and how it can help us with our problems. But to solve some types of problems, it is more interesting to go in the opposite direction… this does not mean being negative, but contrapositive.
Let’s understand then how we can be contrapositive.
Let’s think about some common relationships first:
- If it’s sunny, I go to the beach;
- If there’s zucchini on sale, I’ll buy zucchini;
- If it’s hot, I’ll take a shower.
In these relationships, there is a condition (we’ll call it P) that needs to occur for the other action (we’ll call it Q) to be performed. So, in a slightly more general notation we have “if P then Q”.
Note that ALWAYS when event P occurs, event Q must also occur. However, nothing prevents the event Q from occurring without P having occurred.
- It wasn’t sunny, but I went to the beach;
- The zucchini wasn’t on sale, but I bought it anyway;
- It wasn’t hot, but I took a shower.
Organizing what I explained in the form of a table (also known as a Truth Table) in which we analyze all the possibilities of P and Q occurring and how this results in our “if P then Q” relationship.
|P||Q||P → Q|
The idea of being contrapositive is to think about the negative of these events. That is, “NO P” and “NO Q”, but in this case, we also need to think about the negative relationship between them, which will be “if NOT Q then NOT P”.
Oops, is it weird now? Why have I inverted what was once the condition to now be the consequence?
Let’s look at the previous statements in their contrapositive forms to better understand why we made this inversion:
- If I don’t go to the beach, it’s because it wasn’t sunny;
- If I don’t buy zucchini, it’s because they didn’t have zucchini on sale;
- If I don’t shower, it’s because it wasn’t hot.
Organizing what I explained in the form of a Truth Table in which we analyze all the possibilities of NOT Q and NOT P occurring and how this results in our “if NOT Q then NOT P” relationship.
|NOT Q||NOT P||NOT Q → NOT P|
Wow! We arrive in the same situations where one event occurs because of the other (but you already suspected that would happen, right?)
An example of how being contrapositive can help us solve problems (especially in mathematics), let’s see the following statement:
- For x ∈ ℤ, prove that if x² is even, then x is even.
Proving this problem directly involves writing x² in the form 2.n with n ∈ ℤ, but then we would need to show that √(2.n) is even, and particularly I wouldn’t want to tackle this square root if it’s not really necessary.
An alternative is to think of the contrapositive of this statement, where
- Q: x² is even
- Q: x is even
- P → Q: if x² is even, then x is even
When writing the negation of P and Q, we have
- NO P: x² is not even
- NO Q: x is not even
- NO Q → NO P: if x is not even, x² is not even
But since we are in the set of Integers, if a number is not even, then it is odd. That is, the contrapositive of that statement is:
- For x ∈ ℤ, prove that if x is odd, then x² is odd.
This proof has already appeared in the post Show, Prove or Demonstrate, that is the question.
See how curious, the two statements at first seem different, but when we are more “contrapositive”, we can see that demonstrating one is equivalent to demonstrating the other. And particularly, demonstrating that if x is odd, then x² is odd, is much smoother.
See how changing the way we look at a problem can greatly reduce the difficulty of solving it?