Mathematical Proof/Appendix/Glossary

__NOTOC__ 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 Back to top 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 Back to top 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 C Back to top 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 D Back to top 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 E Back to top Element One of the objects in a set. Notation Equivalent See Logically Equivalent. F Back to top 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 Back to top 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 Back to top 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 Back to top Negation The opposite of a truth statement. The negation of true is false and vice-versa. Logical Reasoning O Back to top Open set A set that is an element of a topology defined on a set Proof by Contradiction R Back to top 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 Back to top Set A collection of items, or elements. Notation Statement See Truth Statement. T Back to top 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