We spent a day or so on Logic and truth tables. It is a big deal that the conditional statement and the contrapositive are logically equivalent. So big of a deal that indirect proofs are based on that fact. So I want my students to see through truth tables that the conditional and contrapositive produce the same truth values.
We start with this example of the Law of Detachment. (A quick Internet search produced no source, and I have no memory about where I got this.)
Law of Detachment: If p →q is a true conditional and p is true, then q is true.
Example: Jordan knows that if he misses the practice the day before the game, then he will not be a starting player in the game. Jordan misses practice on Tuesday, so he concludes that he will not start in the game on Wednesday.
After an example of the Law of Syllogism (which could otherwise be known as the transitive property for conditional statements), we used a few Venn diagrams with conditional statements. We did one together.
And then I let students move around the text to match the given conditionals.
Is the conclusion valid? Or can you find a counterexample?
Then we moved to a brief introduction of symbolic logic. We created truth tables for and, or, and implies. We talked about the difference between the exclusive or and or. (If you ask my daughter whether she wants chocolate cake OR strawberry pie, she will always answer both.)
We talked about the truth values for implies. Why is it that if p is true and q is false, we say that p→q is false, but if p is false and q is true, we say that p→q is true? I remember having a hard time grasping this concept in college – and my professor used an example of it raining purple cats and dogs. Which I didn’t get. It’s taken me years to get this, but I think I finally do get it. I used the example from the Law of Detachment. The team has a rule that if you miss practice the day before the game, you don’t get to start. We’re evaluating the validity of that statement.
The first row is the Law of Detachment. Jordan misses practice. Jordan doesn’t start. The conditional (team rule) is true.
The second row says that Jordan misses practice, but the conclusion is false. Jordan does get to start. If Jordan misses practice and does get to start in the game, then the conditional statement (team rule) is really false.
The third row says that Jordan didn’t miss practice and doesn’t start in the game. Which is fine. And doesn’t contradict the conditional statement (team rule). So we get a true outcome.
The fourth row saw that Jordan didn’t miss practice and does get to start in the game. Which is also fine. And doesn’t contradict the conditional statement (team rule). So we get a true outcome.
After practicing a few truth tables, we created truth tables for an original statement, inverse, converse, and contrapositive.
What do you notice? They each have 3 trues and 1 false. What else do you notice? (Long pauses ensue.) Finally someone notices. The outcomes for the original statement and the contrapositive are the same. The outcomes for the inverse and the converse are the same. Which is a big deal.
The original statement and the contrapositive are logically equivalent. The inverse and the converse are logically equivalent.
We teach students that to prove a statement indirectly, they should assume the conclusion is false and show that there is a contradiction. I want my students to know why this works.
And then of course it would be nice for them to transfer this information not only to proofs but also to standardized test type questions.
Important? I think so, but I guess I am not sure. I don’t know how many of these students will go on to major in something where they need to use indirect proofs. I just know that I wish I had some idea about why indirect proofs worked when I was in school. Either way, the journey continues …