# Mathematical Proof/Appendix/Glossary

This glossary is mostly just for a quick reminder of terms learned in the book and is not meant to be comprehensive or rigorous. Please visit Wikipedia or Wiktionary for more detail.

## A

edit- Arithmetic
- The science of addition and multiplication (subtraction and division are included, since they are the inverse operations of addition and multiplication). Proof by Contrapositive

- Axiom
- A self-evident truth. It is the foundation of logical reasoning. A statement that is accepted as true without proof, which may be assumed in proving that other things are true.Notation

## B

edit- Basis
- A collection of open sets in a set such that the intersection of any two open sets in contains a set Proof by Contradiction

## C

edit- Closed set
- The complement of an open set in a topological space. Proof by Contradiction

- Conclusion
- The result of a given conditional statement. (The "then" clause of a theorem.) This is also sometimes referred to as the
*result*. Constructive Proof

- Conditional statement
- An "if" or an "only-if" statement. It is conditional because its truth value is determined by the truth value of two other statements. Logical Reasoning

- Contrapositive
- The converse and negation of a conditional. The contrapositive of is . Logical Reasoning

- Converse
- The "reverse" of a conditional statement. The converse of is . Logical Reasoning

- Corollary
- That which follows, usually without any necessary argument, from a given result. Constructive Proof

## D

edit- Divisor
- See factor.

- Divide
- An integer
*n*divides an integer*m*, if*n*is a factor of*m*, equivalently, if*m*is a multiple of*n*, or, equivalently, if there's a integer*k*such that . Proof by Contrapositive

## E

edit- Element
- One of the objects in a
*set.*Notation

- Equivalent
- See Logically Equivalent.

## F

edit- Factor
- An integer that divides a given integer. (e.g. 3 is a factor of 6.) This is the "opposite" of multiple. Proof by Contrapositive

## L

edit- Lemma
- A result whose proof is fairly simple or one that is used to simplify or break down a larger argument. Constructive Proof

- Logcially Equivalent
- Two statements that are simultaneously true or simultaneously false are logically equivalent. Logical Reasoning

## M

edit- Multiple
- An integer obtained by multiplying two integers together. (e.g. 4 is a mulitple of 2). This is the "opposite" of factor. Proof by Contrapositive

## N

edit- Negation
- The opposite of a truth statement. The negation of
`true`

is`false`

and vice-versa. Logical Reasoning

## O

edit- Open set
- A set that is an element of a topology defined on a set Proof by Contradiction

## R

edit- Result
- A lemma, theorem, or corollary. A statement of "if-then" that has been proven to be true. Also, the conclusion of such a statement. Constructive Proof

## S

edit- Set
- A collection of items, or
*elements*. Notation

- Statement
- See Truth Statement.

## T

edit- Theorem
- A main result. Usually the proof is somewhat involved and the result is interesting and useful. Constructive Proof

- Topological Space
- A set together with a topology that satisfy the topology axioms. Proof by Contradiction

- Topology
- A collection of subsets of a given set that satisfy the topology axioms. Proof by Contradiction

- Truth Statement
- A statement whose truth value can be determined. Therefore, it is either true or false. Logical Reasoning

- Truth Value
- The assessment of whether a statement is true or false. Logical Reasoning