Plane Geometry - Bruce E. Shapiro

26 SECTION 6. REAL NUMBERSThe intersection of two sets S and T is given byS ∩ T = {x|x ∈ S ∨ x ∈ T }We will use the notation A − B to indicate set difference, which we reada “A minus B”A − B = {x|(x ∈ A) ∧ (x ∉ B)}Venema (and some other texts) use the notation A B for this set.The symbol ∅ represents the empty set. The symbol Q represents the setof all rational numbers.A rational number r is a quotient of two integers p and q, wherer = p/qThe symbol R represents the set of all real numbers. We will not give adefinition of real numbers, but example 5.1 shows that there are numbersthat are not rational. Any real number that cannot be expressed as arational number is called an irrational number. We will see later thatthere is a one-to-one correspondence between the points on a line and thereal numbers, so in a sense, the real numbers give us anything we canmeasure.Axiom 6.1 (Trichotomy of the Real Numbers) Let x, y ∈ R. Thenexactly one of the following is true:x < y, x = y, or x > yAxiom 6.2 (Density) Let x < y ∈ R. Then both of the following aretrue:1. There exists a rational number q such that x < q < y2. There exists an irrational number z such that x < z < yCorollary 6.3 There is an irrational number between any two rationalnumbers.Corollary 6.4 There is a rational number between any two irrational numbers.Theorem 6.5 (Comparison Theorem) Suppose that x, y ∈ R satisfy1. For every rational number q < x, q < y2. For every rational number q < y, q < xthen x = y.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012