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.

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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


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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

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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   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

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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  . Proof by Contrapositive

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

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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

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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

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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

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

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

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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

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


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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   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