Let’s say your best friend is planning to ask your latest crush to the dance, and it’s making you crazy. You’re not even sure if she knows how much you like him, but the whole thing is filling you with anger and jealousy.

You’re starting to see her as the enemy, which is also heartbreaking, because you’ve been friends for like 10 years!

Well, just hold on for a minute: Let’s organize the facts. You said you’re not sure if she knows how much you like him. After all, if she does know, then it might be time to find a new best friend, because she doesn’t care about your feelings. However, if she doesn’t know how much you like him (because you’ve been too embarrassed to admit it, perhaps?), then she’s not the enemy at all! These are two different situations entirely, and you now realize that you need more information before you jeopardize a lifelong friendship.

It’s really helpful to take the information we know and rewrite it into “if . . . then” statements, especially when the “if . . . then” logic is sort of hidden! These are called conditional statements.

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Let’s practice writing conditional statements into “if . . . then” form:

For the breaking nails example, you might have been tempted to say, “If my nails break, then I travel,” because that’s the order the words were first written in. But of course that sentence doesn’t have the same meaning as the original (and it’s a pretty weird statement anyway). The cause should go after the “if,” and the effect should go after the “then.” Make sense?

 

Inductive vs. Deductive Reasoning

If your sister has been late for school every day this year, then you might guess that “your sister will always be late for school.” Or if you look at a bunch of prime numbers, like 3, 7, 19, and 41, you might try to conclude that “all prime numbers are odd,” which is totally false!

 

Inductive reasoning is when we make educated guesses (called conjectures) based on a bunch of specific examples . . . but we haven’t proven anything. Maybe tomorrow your sister’s alarm will work and she’ll end up being on time. And after all, 2 is a prime number, and the last time I checked, 2 wasn’t odd.

 

Your sister being on time to school and the number 2 are counterexamples to the above conjectures. Coming up with counterexamples is like saying, “Ha ha, you’re wrong, and here’s the example to show how wrong you are.” Look, inductive reasoning can be totally useful for understanding patterns, but it’s a bad idea if you’re trying to prove something, because a counterexample could always be lurking around the corner.

Deductive reasoning, on the other hand, means that we are told some general facts, called Givens, and we make specific conclusions based on them with a chain of “if . . . then” logic. This kind of reasoning is airtight (which means there could never be any counterexamples), and it’s what we’ll use when we do proofs.

 

For example, if we are given the facts “All baby ducks are cute” and “Annie loves all cute things,” we can now prove that Annie loves baby ducks. First, we’ll write both statements in “if . . . then” form.

Given: If it’s a baby duck, then it’s cute.
Given: If it’s cute, then Annie loves it.

Therefore: If it’s a baby duck, then Annie loves it.

Ta-da! We just did our first proof. That wasn’t so bad now, was it? By the way, even if you don’t agree that all baby ducks are cute (who are you?), because it was a “Given,” you have to act as if it’s true for the sake of this problem. Let’s make sure we remember the difference between induction and deduction, in case anyone asks (like on a test).

 

Remember, the key to deduction is that we can only conclude things that must be true from the Givens—not things that might be true. Let’s practice!

Given the statements below, use deduction to reach a new conclusion, if any. I’ll do the first one for you.

1. Given: Bonzo has a moustache. All monkeys have moustaches.

 

Working out the solution: In this case, it might seem like we could conclude that Bonzo is a monkey, but we can’t! You see, with this information, there’s no airtight argument that leads us from A to B. I mean, Bonzo might be a monkey, but he might also be a giraffe who happens to have a moustache. See what I mean?

Answer: No conclusion possible

2. Given: All Barbies are dolls. All dolls are creepy.
3. Given: All puppies like bones. Sparky likes bones.
4. Given: All fruit grows on trees. All apples are fruit.
5. Given: All aliens speak Martian. Debbie speaks Martian.

(Answers at bottom)

 

These types of deduction problems show up a lot on standardized tests. If you struggled with these (this is totally normal!), just hang in there and give yourself a break; you’re training your brain in a whole new way. Remember the “if . . . then” statements we did a few pages ago? We’ll soon be combining those with deduction . . . and smelly socks.

The Converse, Inverse, and Contrapositive of a Statement

Let’s say we’re given All hot guys wear smelly socks. We could rewrite this as “If a guy is hot, then he wears smelly socks.” Great. Now, if we’re told that some guy (Zac) wears smelly socks, must he be hot? Nope! All sorts of people wear smelly socks, not just hot guys, after all. So the statement, “If a guy wears smelly socks, then he’s hot,” isn’t necessarily true. In fact, we’ve swapped the cause and effect in the statement, and that’s called the converse.

The converse, inverse, and contrapositive of a statement are all ways to change an “if . . . then” statement by moving around its “cause and effect” parts in very specific ways.

 

Let’s use p’s and q’s to create generic “if . . . then” statements, just to make everything shorter and easier to see. Later, we can substitute phrases like “has smelly socks” if we want.

 

Original statement: If p, then q.
Converse: If q, then p.
Inverse: If not p, then not q.
Contrapositive: If not q, then not p.

 

Believe it or not, the contrapositive actually tells us the same information as the original statement. I mean, if a guy doesn’t wear smelly socks, then we know he can’t be hot . . . because if he were hot, then he’d be wearing smelly socks!

Same information, written differently!

Original statement ⇔ Contrapositive

 

In our particular case, the p is hotness and the q is smelly socks, right?