Axioms for the Real Numbers and Integers

Axioms for the Real numbers http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/Lectures/L5.html

⊛ ⊙ ⊚ ⊗ ⊕ ⊘ ⊜ ⊖ ⊡ ⊞ ⊞ ⊠ ⊡ ⋅ ⊹ ⋅ ⋆ ⋄ ⋇

1. Existence

   • there exists a set ℝ containing all real numbers, ∃ℝ = {reals}

   • ℝ contains the entire set of integers, ℤ ⊆ ℝ

   • inclusion relation of number sets: ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ ⊆ ℍ

2. Closure

   • the set ℝ is closed under addition and multiplication:

   • ∀a,∀b. a,b ∈ ℝ -> a ⊚ b ∈ ℝ, where ⊚ is + or *

3. Commutativity

   • ∀a,∀b. a,b ∈ ℝ -> a ⊚ b = b ⊚ a, where ⊚ is + or *

4. Associativity

   • ∀a,∀b,∀c. (a,b,c ∈ ℝ) -> (a ⋇ b) ⋇ c = a ⋇ (b ⋇ c) = a ⋇ b ⋇ c, where ⋇ is + or *

5. Distributivity

   • left : a ⋄ (b ⊕ c) = (a ⋄ b) ⊕ (a ⋄ c)

   • right: (a ⊕ b) ⋄ c = (a ⋄ c) ⊕ (b ⋄ c)

6. Zero: 0 is an integer that satisfies a + 0 = a = 0 + a for every real number a.

7. One is an element that satisfies: ∀a ∈ ℝ . a · 1 = a = 1 · a

8. Additive inverses: If a is any real number, there is a unique real number −a such that a + (−a) = 0. If a is an integer, then so is −a.

9. Multiplicative inverses: If a is any nonzero real number, there is a unique real number a−1 such that a · a−1 = 1.

10. Trichotomy: If a and b are real numbers, then one and only one of the following 3 statements is true: a < b, a = b, or a > b.

11. Closure of R+: If a and b are positive real numbers, then so are a + b and ab

12. Addition law for inequalities: If a, b, and c are real numbers and a < b, then a + c < b + c

13. The well ordering axiom: Every nonempty set of positive integers contains a smallest integer.

14. The least upper bound axiom: Every nonempty set of real numbers that has an upper bound has a least upper bound.

Properties of Operations

These theorems can be proved from the axioms in the order listed below.

1. Properties of zero

   • a) a − a = 0

   • b) 0 − a = −a

   • c) 0 · a = 0

   • d) If ab = 0, then a = 0 or b = 0

2. Properties of signs

   • a) −0 = 0

   • b) −(−a) = a

   • c) (−a)b = −(ab) = a(−b)

   • d) (−a)(−b) = ab

   • e) −a = (−1)a

3. More distributive properties

   • a) −(a + b) = (−a) + (−b) = −a − b

   • b) −(a − b) = b − a

   • c) −(−a − b) = a + b

   • d) a + a = 2a

   • e) a(b − c) = ab − ac = (b − c)a

   • f) (a + b)(c + d) = ac + ad + bc + bd

   • g) (a + b)(c − d) = ac − ad + bc − bd = (c − d)(a + b)

   • h) (a − b)(c − d) = ac − ad − bc + bd

4. Properties of inverses (a) If a is nonzero, then so is a−1 (b) 1−1 = 1 (c) (a−1)−1 = a if a is nonzero (d) (−a)−1 = −(a−1) if a is nonzero (e) (ab)−1 = a−1b−1 if a and b are nonzero (f) (a/b)−1 = b/a if a and b are nonzero

5. Properties of quotients (a) a/1 = a. (b) 1/a = a−1 if a is nonzero. (c) a/a = 1 if a is nonzero. (d) (a/b)(c/d) = (ac)/(bd) if b and d are nonzero. (e) (a/b)/(c/d) = (ad)/(bc) if b, c, and d are nonzero. (f) (ac)/(bc) = a/b if b and c are nonzero. (g) a(b/c) = (ab)/c if c is nonzero. (h) (ab)/b = a if b is nonzero. (i) (−a)/b = −(a/b) = a/(−b) if b is nonzero. (j) (−a)/(−b) = a/b if b is nonzero. (k) a/b + c/d = (ad + bc)/(bd) if b and d are nonzero. (l) a/b − c/d = (ad − bc)/(bd) if b and d are nonzero

6. Transitivity of inequalities (a) If a < b and b < c, then a < c (b) If a ≤ b and b < c, then a < c (c) If a < b and b ≤ c, then a < c (d) If a ≤ b and b ≤ c, then a ≤ c

7. Other Properties of inequalities (a) If a ≤ b and b ≤ a, then a = b (b) If a < b, then −a > −b (c) 0 < 1 (d) If a > 0, then a−1 > 0 (e) If a < 0, then a−1 < 0 (f) If a < b and a and b are both positive, then a−1 > b−1. (g) If a < b and c < d, then a + c < b + d. (h) If a ≤ b and c < d, then a + c < b + d. (i) If a ≤ b and c ≤ d, then a + c ≤ b + d. (j) If a < b and c > 0, then ac < bc. (k) If a < b and c < 0, then ac > bc. (l) If a ≤ b and c > 0, then ac ≤ bc. (m) If a ≤ b and c < 0, then ac ≥ bc. (n) If a < b and c < d, and a, b, c, d are nonnegative, then ac < bd. (o) If a ≤ b and c ≤ d, and a, b, c, d are nonnegative, then ac ≤ bd. (p) ab > 0 iff a and b are both positive or both negative. (q) ab < 0 iff one is positive and the other is negative. (r) There is no smallest positive real number. (s) (Density) If a and b are two distinct real numbers, then there are infinitely many rational numbers and infinitely many irrational numbers between a and b

8. Properties of squares (a) For every a, a2 ≥ 0 (b) a2 = 0 iff a = 0 (c) a2 > 0 iff a > 0 (d) (−a)2 = a2 (e) (a−1)2 = 1/a2 (f) If a2 = b2, then a = ±b (g) If a < b and a and b are both nonnegative, then a2 < b2 (h) If a < b and a and b are both negative, then a2 > b2

9. Properties of Square Roots (a) If a is any nonnegative real number, there is a unique nonnegative real number √a such that √a
   2 = a (b) If a = b and a and b are both nonnegative, then √a = √b (c) If a < b and a and b are both nonnegative, then √a < √b (d) If a2 = b and b is nonnegative, then a = ±√b

10. Properties of Absolute Values

    • a) If a is any real number, then |a| ≥ 0.

    • b) |a| = 0 iff a = 0.

    • c) |a| > 0 iff a 6= 0.

    • d) | − a| = |a|.

    • e) |a| = √a2.

    • f) |a| = max{a, −a}.

    • g) |a−1| = 1/|a| if a 6= 0.

    • h) |ab| = |a| |b|.

    • i) |a/b| = |a|/|b| if b 6= 0.

    • j) |a| = |b| iff a = ±b.

    • k) If a and b are both nonnegative, then |a| ≥ |b| iff a ≥ b.

    • l) If a and b are both negative, then |a| ≥ |b| iff a ≤ b.

    • m) (The triangle inequality) |a + b| ≤ |a| + |b|.

    • n) (The reverse triangle inequality) |a| − |b| ≤ |a − b|

11. Order properties of integers

    • a) 1 is the smallest positive integer.

    • b) If m and n are integers such that m > n, then m ≥ n + 1

    • c) There is no largest or smallest integer

12. Properties of Even and Odd Integers

    • In each of the following statements, m and n are assumed to be integers

    • a) n is even iff n = 2k for some int k,

      and odd iff n = 2k + 1 for some int k

    • b) m + n is even iff m and n are both odd or both even

    • c) m + n is odd iff one of the summands is even and the other is odd

    • d) mn is even iff m or n is even

    • e) mn is odd iff m and n are both odd

    • f) n2 is even iff n is even, and odd iff n is odd

13. Properties of Exponents

    • In these statements, m and n are positive integers

    • a) anbn = (ab)n

    • b) am+n = ambn

    • c) (am)n = amn

    • d) an/bn = (a/b)n if b is nonzero