# Logic and Proof

### Terms

### 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."### Detachment

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

*and*the conditional statement are true.

### 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.### 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.### 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. In this case, we can multiply both sides by 0, but it will just give us 0 = 0.### 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.

### 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 going 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*.

### Symmetrical 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.

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