# Mathematical Proof/Appendix/Glossary

A | B | C | D | E | F | L | M | N | O | R | S | T

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.

## AEdit

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

## BEdit

Basis
A collection $\mathcal B$ of open sets in a set $X$ such that the intersection of any two open sets in $X$ contains a set $B\in \mathcal B.$ Proof by Contradiction

## CEdit

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 statments. Logical Reasoning
Contrapositive
The converse and negation of a conditional. The contrapositive of $P\Rightarrow Q$ is $\lnot Q \Rightarrow \lnot P$. Logical Reasoning
Converse
The "reverse" of a conditional statement. The converse of $P \Rightarrow Q$ is $P \Leftarrow Q$. Logical Reasoning
Corollary
That which follows, usually without any necessary argument, from a given result. Constructive Proof

## DEdit

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). Proof by Contrapositive

## EEdit

Element
One of the objects in a set. Notation
Equivalent
See Logically Equivalent.

## FEdit

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

## LEdit

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

## MEdit

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

## NEdit

Negation
The opposite of a truth statement. The negation of true is false and vice-versa. Logical Reasoning

## OEdit

Open set
A set that is an element of a topology $\tau$ defined on a set $X.$ Proof by Contradiction

## REdit

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

## SEdit

Set
A collection of items, or elements. Notation
Statement
See Truth Statement.

## TEdit

A set $X$ together with a topology $\tau$ that satisfy the topology axioms. Proof by Contradiction