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.

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   is   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