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.

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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 \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

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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 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

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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). 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 \tau defined on a set X. 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 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
Last modified on 7 October 2006, at 02:52