# Mathematical Proof/Appendix/Glossary

* Mathematical Proof
* Introduction
* Logical Reasoning
* Notation
* Methods of Proof
* Constructive Proof
* Proof by Contrapositive
* Proof by Contradiction
* Proof by Induction
* Counterexamples
* Appendix
* Answer Key
* Symbols Used in this Book
* 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

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### 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

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### 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

## C

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### 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 $P\Rightarrow Q$ 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

## D

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### 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 $n*k=m$. Proof by Contrapositive

## E

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### Element

    One of the objects in a set. Notation

### Equivalent

    See Logically Equivalent.

## F

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### 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

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### 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

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### 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

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### Negation

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

## O

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### Open set

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

## R

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### 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

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### Set

    A collection of items, or elements. Notation

### Statement

    See Truth Statement.

## T

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### Theorem

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

### Topological Space

    A set $X$ together with a topology $\tau$ 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