MAT 1348 Study Guide - Final Guide: Rational Number, Subset, Idempotence
17 May 2020
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Theorem: if 0 > 1, then 2 is a rational number. Theorem: if 0 < 1, then 4 is a rational number. Theorem: let a and b be real numbers. Then the following are equivalent: a < b, a < a + b. Proof: when showing the equivalence of more than two. It su ces to statements, one can go in a circular fashion. prove that 1 2, 2 3 and 3 1. (why?) Theorem: the equation x3 + x + 1 = 0 has no rational roots (i. e. , no rational solutions for x). Proof (combination of proof by contradiction and proof by cases): Defn: a set is a collection of distinct objects; these objects are then called elements of the set. A set can have a nite (even zero) or an in nite number of elements. Conventionally we use capital letters to denote sets. To express that an element x is in a set a, we write x a.
