General Structure of a Theorem The purpose of a theorem is to present a mathematical truth. Of course, just stating that something is true does not make it so. Mathematics history is littered with “theorems” that later were discovered to be false. What that means is the original proof was faulty. Our goal is to develop proofs that are solid. This sheet attempts to show how to minimize the chance of error. A theorem has two parts: its statement and its proof. The Theorem Statement is subdivided into two constituents: Hypotheses (these are statements that are assumed to be true without question; there is no option for believing them or not) Conclusions (these are statements of what is to be proven) It is important to be able to identify the hypotheses and conclusions in a theorem statement. The proof is a series of statements leading the reader to incontrovertible agreement that the conclusions are true under the assumption that the hypotheses are true. There are various types of statements in the
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