Mathematical Proof/Appendix/Glossary

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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