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Unformatted text preview: Math 280 Strategies of Proof Fall 2007 Class Notes 1.1 Statements and the Logical Connectives The study of an area of mathematics begins by first defining the mathematical objects to be studied. For instance, in plane geometry, the objects in question include points, lines, circles, triangles, and parallel lines. In linear algebra, the objects include vector spaces, subspaces, linear transformations, and eigenvalues. In abstract algebra, the objects include groups, rings, fields, modules, and algebras. After making these definitions, the mathematician then proceeds to make statements about the mathematical objects. There are two basic types of mathematical statements: axioms and theorems. Axioms are statements that are assumed to be true. In the past, axioms were considered statements so obvious that their truth was self-evident. For instance, the ancient Greek mathematicians considered the statement that through any two distinct points, there passes exactly one line to be an axiom of geometry. This type of thinking assumes that the mathematical objects, in this case, points and lines, exist in the physical world. In modern mathematics, the mathematical objects in question are often abstract entities which need not have any connection to the physical world. The axioms in this modern viewpoint are then just assumptions that are part of the definitions of the mathematical structures under examination. For instance, a group is defined in abstract algebra to be a nonempty set of objects together with a binary operation, or rule, for combining pairs of objects from the set, which satisfies four properties: closure, associativity, the existence of an identity element, and the existence of inverses. The set may consist of objects from the physical world or may be purely abstract. The four properties in the definition are the axioms of the group. Even though the objects in question may be purely abstract, mathematicians still refer to the axioms as being true. The other type of mathematical statement is a theorem. Lemmas and corollaries are special types of theorems, but theorems none the less. In the past, mathematicians considered theorems to also be true statements. However, the truth of a theorem was not self-evident as was the truth of an axiom. Instead, its truth had to be deduced from the axioms using the laws of logic. A proof of a theorem was a valid logical argument verifying the truth of a theorem. For instance, the statement from geometry The sum of the angles of a triangle equals 180 . is a theorem rather than an axiom since its truth is not self-evident. In the modern point of view, a theorem is still a statement which is a logical consequence of the axioms. A proof is then a valid logical argument demonstrating that the theorem does indeed follow from the axioms, as opposed to demonstrating the truth of the theorem. Even though it may not be accurate to refer to the truth of a theorem (or of an axiom), mathematicians today nevertheless...
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