Converse, Inverse, and Contrapositive
In the last topic, we discussed conditional statements and how to negate them. Here we present a few additional ways of altering the usual \(p\rightarrow q\) structure.
In other words, if given a conditional statement \(p\rightarrow q\), we can find the converse \(q\rightarrow p\) simply by swapping the order of the statements. You can think of the converse as the “reverse” of a conditional statement.
Examples of Converse
| Original Implication \(p\rightarrow q\) | Converse (Implication) \(q\rightarrow p\) |
| “If I eat paint chips, then I am not okay” | “If I am not okay, then I eat paint chips” |
| “If you bought the top and sold the bottom, then you lost money” | “If you lost money, then you bought the top and sold the bottom” |
| “If at first you don’t succeed, then you shouldn’t try skydiving” | “If you shouldn’t try skydiving, then you don’t succeed at first” |
In other words, the inverse of a conditional statement can be found by negating both \(p\) and \(q\) without changing the order of the implication.
Examples of Inversion
| Original Implication \(p\rightarrow q\) | Inverse (Implication) \(\neg p \rightarrow \neg q\) |
| “If I am happy, then I am healthy” | “If I am unhappy, then I am not healthy” |
| “If I smile nonstop, then people will avoid me” | “If I don’t smile nonstop, then people won’t avoid me” |
| “If I don’t shave my mustache, then I look like a catfish” | “If I shave my mustache, then I don’t look like a catfish” |
In other words, to find the contrapositive of the implication \(p\rightarrow q\), first negate both \(p\) and \(q\), and then “flip” the statement so you have \(\neg q\) first and \(\neg p\) second in the implication.
Examples of Contrapositive
| Original Implication \(p\rightarrow q\) | Contrapositive (Implication) \(\neg q \rightarrow \neg p\) |
| “If I wear a mask, then I can’t smell your B.O.” | “If I can smell your B.O., then I am not wearing a mask” |
| “If I think bran flakes are a tasty cereal, then I must be getting old” | “If I am not getting old, then I don’t think bran flakes are a tasty cereal” |
| “If I work without ceasing, then I won’t have time to enjoy life.” | “If I have time to enjoy life, then I don’t work without ceasing” |
| “If \(x<0\) then \(x\) is negative” | “If \(x\) is nonnegative, then \(x\geq 0\)” |
Bringing These All Together
You may have noticed in the last table for contrapositive that the two entries in each row say essentially the same thing, or rather, have the same meaning. This is because of the following key idea.
In the examples at the end of this topic you can prove this theorem for yourself by generating a truth table for the implication \(p\rightarrow q\), its converse, inverse, and contrapositive, and then comparing the columns of each.
“Only If”
Consider the following statement
“I will sleep only if I get this lesson written”
This feels like a conditional statement, and we’d like to write it as one.
Note that what this statement is effectively saying is “the only possible way I will sleep is if I get this lesson written.” So the statement above is equivalent to “If I sleep, then I got this lesson written” because it must be the case that I got the lesson written if I am found sleeping!
Another way of thinking about “\(p\) only if \(q\)” is: “the only possible way that \(p\) is true (or occurs) is if \(q\) is true (or occurs)” or “If \(p\) is true, then it must have been the case that \(q\).” In other words, it is necessary for \(q\) to be true if \(p\) is found to be true.
In other words, \(p\) is a sufficient condition for \(q\) if \(q\) comes as a result of \(p\); i.e. if \(p\) is true then it follows that \(q\) must also be true. \(p\) is a necessary condition for \(q\) if it is required that \(p\) be true in order for \(q\) to also be true; i.e. the truth of \(q\) implies the truth of \(p\); i.e. \(q\rightarrow p\).
Quick Examples Converting From “Necessary/Sufficient” to Conditional Statements
In English, when phrases use terms like “sufficient that” or “it is necessary that” we can convert these phrases to their conditional form based on the definitions of “Necessary” and “Sufficient” as seen above. The table below indicates how this can be done
| “Necessary/Sufficient” Version of a Phrase | Conditional Version of the Phrase |
| “Doing one’s homework is necessary for earning good grades” | “One earns good grades only if they do their homework” |
| “To stay on top of my to-do list, it is necessary that I work more than 90 hours per week” | “I will stay on top of my to-do list only if I work more than 90 hours per week. |
| “A number \(n\) being divisible by \(2\) is a sufficient condition for \(n\) to be an even number” | “If \(n\) is divisible by \(2\), then \(n\) is even” |
| “Getting regular exercise is a sufficient condition for living longer” | “If one gets regular exercise, then they will (likely) live longer.” |
Biconditionals: “If and Only If”
Recalling a conversation I’ve had with my wife recently, I stated:
“I will do the dishes if, and only if, you scoop the cat litter”
I hate doing the litter and she hates doing the dishes. Thinking very carefully about what this statement means, the only way I am doing the dishes is if my wife scoops the cat litter, and the only way she does the litter is if I do the dishes. If one part of the deal is upheld, the other must be as well. Quid pro quo.
Just like in the cat litter-dishes example above, the biconditional statement \(p\leftrightarrow q\) implies that \(p\) is true precisely when \(q\) is true (and vice versa), and \(p\) is false precisely when \(q\) is false (and vice versa). Biconditional statements set up a form of equivalence between statements \(p\) and \(q\). The statement \(p\leftrightarrow q\) is true provided that \(p\) and \(q\) are either simultaneously true or simultaneously false.
The last definition and the discussion regarding necessary and sufficient conditions prompt the following key idea.
- “If I want good produce, then I should grow my own vegetables”
- “If I water my plants, then vegetables will grow”
- “If I over-water my plants, then leaves turn yellow”
- “If that rabbit eats my vegetables again, then it’s rabbit stew for dinner!”
| Original Implication \(p\rightarrow q\) | Converse Implication \(q\rightarrow p\) | Inverse Implication \(\neg p \rightarrow \neg q\) | Contrapositive Implication \(\neg q \rightarrow \neg p\) |
| “If I want good produce, then I should grow my own vegetables” | “If I should grow my own vegetables, then I want good produce” | “If I don’t want good produce, then I should not grow my own veggies” | “If I shouldn’t grow my own veggies, then I don’t want good produce” |
| “If I water my plants, then vegetables will grow” | “If vegetables will grow, then I water my plants” | “If I don’t water my plants, then vegetables won’t grow” | “If vegetables don’t grow, then I don’t water my plants” |
| “If I over-water my plants, then leaves turn yellow” | “If leaves turn yellow, then I over-water my plants” | “If I don’t over-water my plants, then leaves don’t turn yellow” | “If leaves don’t turn yellow, then I don’t over-water my plants” |
| “If that rabbit eats my vegetables again, then it’s rabbit stew for dinner!” | “If it’s rabbit stew for dinner, then that rabbit ate my veggies again.” | “If that rabbit does not eat my vegetables again, then it’s not rabbit stew for dinner” | “If it’s not rabbit stew for dinner, then that rabbit didn’t eat my vegetables again” |
Suppose we have a conditional statement \(p \rightarrow q\) such as “If I want good produce, then I should grow my own vegetables” where \(p\) is “I want good produce” and \(q\) is “I should grow my own vegetables.”
To find the converse, simply swap the order of \(p\) and \(q\) in the original implication
To find the inverse, simply negate both \(p\) and \(q\) without changing their locations in the conditional statement.
To find the contrapositive, negate \(p\) and \(q\) AND swap their locations in the conditional statement. Contrapositive is the same thing as the converse of the inverse.
- “To retire comfortably today, it is necessary to save $1.6 million.”
- “It is necessary to pet your cat daily if you want to keep them happy.”
- “If you want to get good at math, it is necessary to work challenging problems”
- “To solve a challenging problem, it is sufficient to focus one’s attention on the problem for many hours.”
- “Moving straight into the sky at 13km/sec is sufficient for leaving the earth’s atmosphere”
- “\(n\) being divisible by \(4\) is sufficient for \(n\) to be even.”
- “One can retire comfortably today only if they save $1.6 million”
- “You will keep your cat happy only if you pet said cat daily”
- “You can get good at math only if you work challenging problems”
- “If one focuses their attention on a problem for many hours, then they will solve the problem.”
- “If something moves straight into the sky at 13km/sec, then it will leave the earth’s atmosphere.”
- “If \(n\) is divisible by \(4\) then \(n\) is even”
“necessary” implies that one thing occurs only if another thing occurs; i.e. one can’t happen without the other.
“sufficient” implies that one condition being met is “good enough” to ensure that the other is also met; hence “sufficient” corresponds to “if… then….”
- Suppose \(n\) is a natural number. For \(n\) to be even, is it necessary or sufficient (or both) that \(4\vert n\)?
- Suppose \(n\) is a natural number. For \(n\) to be even, is it necessary or sufficient (or both) that \(2\vert n\)?
- Suppose \(x\in \mathbb{R}\). For \(x<5\) is it necessary or sufficient (or both) that \(x\) is negative?
- Suppose \(n\) is a natural number. For \(\vert 6|n\), is it necessary or sufficient (or both) that \(2\vert n\)?
- Suppose \(x\in \mathbb{R}\). For \(x\) to be nonnegative, is it necessary or sufficient (or both) that \(x>-10\)?
- For \(n\) to be even, it is sufficient that \(4\vert n\), but not necessary. Not all even numbers are divisible by \(4\), so that doesn’t need to be true for \(n\) to be even.
- For \(n\) to be even, is it necessary and sufficient that \(2\vert n\). All even numbers are divisible by \(2\) and any number divisible by \(2\) must, by definition, be even. So, \(n\) is even \(\leftrightarrow\) \(2\vert n\) (or \(n\) is even iff \(2\vert n\).
- For \(x<5\) it is sufficient for \(x\) to be negative (i.e. if \(x\) is negative, then certainly \(x<5\)), but its not necessary. Note that \(x=4\) is less than \(5\) but is not negative.
- For \(6\vert n\), it is necessary that \(2\vert n\). Any number divisible by \(6\) must be divisible by \(2\) and \(3\), by definition. Yet, it is not sufficient that just \(2\vert n\) for numbers divisible by \(6\); that is, not just any old number divisible by \(2\) is also divisible by \(6\). However, to compare, the condition “\(2\vert n\) AND \(3\vert n\)” would be a sufficient condition for \(6\vert n\).
- For \(x\) to be nonnegative, it is necessary that \(x>-10\), because you can’t have a nonnegative number that is less than \(-10\)! It is NOT sufficient that \(x>-10\) for \(x\) to be nonnegative. There is certainly a number \(x\) greater than \(-10\) such that \(x\) is negative. Try finding one!