# Logic and Proof Terms

## Get down with the lingo

### Addition Property

We can add the same number to both sides of an equation without causing harm to the equation or ourselves. Not true if we're trying to add a fork to an electrical outlet.### Angle Bisector

A line segment that splits an angle in half so that it creates two congruent angles.### Axiom

See "Postulate."### Base Case

The first step in an inductive proof. We take the smallest or simplest value in our set, usually*n*= 1, and prove that our general statement is true with that value. Also a great name for a rap group that specializes in inductive proofs. "Ladies and gents, please welcome to the stage…

*Base Case*." (Cue wild applause.)

### Conditional Statement

A compound proposition that says*if*the hypothesis is true,

*then*the conclusion will be true. It's surprisingly popular for a statement with so many conditions for hanging out. In symbols, it's

*p*→

*q*.

### Congruent

Exactly equal in measure or identical. #twinsies### Conjunction

Two statements joined together by the word "and." Both parts of the conjunction must be true for the entire statement to be true. That's the truth,*and*don't you forget it.

### Contrapositive

A conditional statement where the hypothesis and conclusion switch places*and*a big "not" is put before both of them: ~

*q*→ ~

*p*, or not

*q*→ not

*p*.

### Converse

A conditional statement where the hypothesis and conclusion switch places. It's like Wife Swap in the math world:*q*→

*p*.

### Counterexample

Any*n*-value that disproves the general statement in an inductive proof. It ruins the entire proof, like a drop of garbage water in a batch of cookies.

### Detachment

The idea that we can separate the conclusion and hypothesis from the conditional statement itself. The law might exist, but it will only be obeyed if there exist people that can obey it*and*it's passed. Just the same, the hypothesis only leads to the conclusion if both the hypothesis

*and*the conditional statement are true.

### Disjunction

Two statements joined together by the word "or." Only one statement has to be true for the disjunction to be true. Or was that some other type of statement? (No, it wasn't.)### Division Property

We can divide both sides of an equation by any number as long as it's not zero. If we divide by zero, we could be blamed for the apocalypse.### Formal Proof

Also called a two-column proof, it's a proof arranged in the form of—you guessed it!—two columns. One lists the statements you're claiming as true and the other lists the reasons for why they're true.### Indirect Proof

A proof that proves that the opposite of what it really wants to prove is false. Basically, it proves a claim by proving that the opposite isn't true. Talk about beating around the bush.### Induction Hypothesis

The second step in an inductive proof. We assume our initial statement is true when*n*=

*k*. Yep, it's the one area in math where you're allowed to say, "It's true because I

*said so*."

### Informal Proof

Also called a paragraph proof, it's a proof that takes the form of—you guessed it!—a paragraph. Think more stream-of-consciousness and less Excel spreadsheet. It's still a proof though, so make sure to give your reasons.### Inverse

A conditional statement where "not" is placed before both the hypothesis and conclusion: ~*p*→ ~

*q*, or not

*p*→ not

*q*.

### Midpoint

A point that bisects a line segment into two congruent segments. It's the halfway point.### Multiplication Property

We can multiply both sides of an equation by the same number without messing with the truth of the equation.### Negation

Putting "not" into a proposition to make it mean the opposite of the original statement. If you get too many negations together, they tend to not not not not not not be annoying.### Postulate

An assumption we make about whatever objects we're talking about that's treated as a fact. "Axiom" is its alter-ego.### Proof

An argument laid out step-by-step or in paragraph form that explains why something is true. Just like the courtroom scenes in*Law and Order*, except with math.

### Proof By Contradiction

A type of proof where we assume that the thing we're trying to prove is*not*true, and then show how that assumption leads to some kind of logical contradiction. Also goes by the nicknames

*indirect proof*and

*Lyin' Larry*.

### Proof By Deduction

A type of top-down reasoning where we start with a general theory or statement, then apply it to a specific example. Sherlock is a big fan.### Proof By Induction

A type of bottom-up reasoning where we start with a specific statement and use it to prove a general theory. First we prove the base case, then we assume the general theory is true when*n*=

*k*, and we wrap things up by proving the theory is true when

*n*=

*k*+ 1.

### Reflexive Property

Simply put, things are exactly what they are. Everything equals itself.*A*=

*A*.

*B*=

*B*. Bubba Gump shrimp = Bubba Gump shrimp. Convincing (and delicious), right?

### Substitution Property

If two things are equal, they are interchangeable. That means one can be substituted for another. How else would we have been able to solve for all those variables in algebra?### Subtraction Property

It's the green light for subtracting the same number from both sides of an equation. It can be 0 or it can be 5 million, but as long as we're doing the same thing to both sides of the equation, it doesn't make a difference.### Syllogism

A shortcut to conditional statements. If*p*→

*q*and

*q*→

*r*and

*r*→

*s*and

*s*→

*t*, instead of going through each one separately, it allows us to say directly that

*p*→

*t*.

### Symmetric Property

This gets to the true concept of equality. If*A*=

*B*, then

*B*=

*A*. As long as the equal sign keeps them apart (how sad!), it doesn't matter what side they land on.

### Theorem

A fact that's already been proven. With theorems, all the work has already been done for us.### Transitive Property

In the simplest terms, if*A*=

*B*and

*B*=

*C*, then

*A*=

*C*. Sort of obvious, but super useful.