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\u2026 this does not mean being negative, but contrapositive.<\/p>\n\n\n\n
Let’s understand then how we can be contrapositive.<\/p>\n\n\n\n
Let’s think about some common relationships first:<\/p>\n\n\n\n
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 \u201cif P then Q\u201d.<\/p>\n\n\n\n
Note that ALWAYS when event P occurs, event Q must also occur. However, nothing prevents the event Q from occurring without P having occurred.<\/p>\n\n\n\n
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 \u201cif P then Q\u201d relationship.<\/p>\n\n\n\n The idea of \u200b\u200bbeing contrapositive is to think about the negative of these events. That is, \u201cNO P\u201d and \u201cNO Q\u201d, but in this case, we also need to think about the negative relationship between them, which will be \u201cif NOT Q then NOT P\u201d.<\/p>\n\n\n\n Oops, is it weird now? Why have I inverted what was once the condition to now be the consequence?<\/p>\n\n\n\n Let’s look at the previous statements in their contrapositive forms to better understand why we made this inversion:<\/p>\n\n\n\n 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 \u201cif NOT Q then NOT P\u201d relationship.<\/p>\n\n\n\n Wow! We arrive in the same situations where one event occurs because of the other (but you already suspected that would happen, right?)<\/p>\n\n\n\n An example of how being contrapositive can help us solve problems (especially in mathematics), let’s see the following statement:<\/p>\n\n\n\n Proving this problem directly involves writing x\u00b2 in the form 2.n with n \u2208 \u2124, but then we would need to show that \u221a(2.n) is even, and particularly I wouldn’t want to tackle this square root if it’s not really necessary.<\/p>\n\n\n\n An alternative is to think of the contrapositive of this statement, where<\/p>\n\n\n\n When writing the negation of P and Q, we have<\/p>\n\n\n\n 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:<\/p>\n\n\n\nP<\/strong><\/td> Q<\/strong><\/td> P \u2192 Q<\/strong><\/td><\/tr> TRUE<\/td> TRUE<\/td> TRUE<\/td><\/tr> TRUE<\/td> FALSE<\/td> FALSE<\/td><\/tr> FALSE<\/td> TRUE<\/td> TRUE<\/td><\/tr> FALSE<\/td> FALSE<\/td> TRUE<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n NOT Q<\/strong><\/td> NOT P<\/strong><\/td> NOT Q \u2192 NOT P<\/strong><\/td><\/tr> FALSE<\/td> FALSE<\/td> TRUE<\/td><\/tr> TRUE<\/td> FALSE<\/td> FALSE<\/td><\/tr> FALSE<\/td> TRUE<\/td> TRUE<\/td><\/tr> TRUE<\/td> TRUE<\/td> TRUE<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n