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

βŠ› βŠ™ ⊚ βŠ— βŠ• ⊘ ⊜ βŠ– ⊑ ⊞ ⊞ ⊠ ⊑ β‹… ⊹ β‹… ⋆ β‹„ ⋇

  1. Existence

    • there exists a set ℝ containing all real numbers, βˆƒβ„ = {reals}

    • ℝ contains the entire set of integers, β„€ βŠ† ℝ

    • inclusion relation of number sets: β„• βŠ† β„€ βŠ† β„š βŠ† ℝ βŠ† β„‚ βŠ† ℍ

  2. Closure

    • the set ℝ is closed under addition and multiplication:

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

  3. Commutativity

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

  4. Associativity

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

  5. Distributivity

    • left : a β‹„ (b βŠ• c) = (a β‹„ b) βŠ• (a β‹„ c)

    • right: (a βŠ• b) β‹„ c = (a β‹„ c) βŠ• (b β‹„ 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: βˆ€a ∈ ℝ . a Β· 1 = a = 1 Β· a

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

  9. Multiplicative inverses: If a is any nonzero real number, there is a unique real number aβˆ’1 such that a Β· aβˆ’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 βˆ’ a = 0

    • b) 0 βˆ’ a = βˆ’a

    • c) 0 Β· a = 0

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

  2. Properties of signs

    • a) βˆ’0 = 0

    • b) βˆ’(βˆ’a) = a

    • c) (βˆ’a)b = βˆ’(ab) = a(βˆ’b)

    • d) (βˆ’a)(βˆ’b) = ab

    • e) βˆ’a = (βˆ’1)a

  3. More distributive properties

    • a) βˆ’(a + b) = (βˆ’a) + (βˆ’b) = βˆ’a βˆ’ b

    • b) βˆ’(a βˆ’ b) = b βˆ’ a

    • c) βˆ’(βˆ’a βˆ’ b) = a + b

    • d) a + a = 2a

    • e) a(b βˆ’ c) = ab βˆ’ ac = (b βˆ’ c)a

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

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

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

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

  5. Properties of quotients (a) a/1 = a. (b) 1/a = aβˆ’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) (βˆ’a)/b = βˆ’(a/b) = a/(βˆ’b) if b is nonzero. (j) (βˆ’a)/(βˆ’b) = a/b if b is nonzero. (k) a/b + c/d = (ad + bc)/(bd) if b and d are nonzero. (l) a/b βˆ’ c/d = (ad βˆ’ 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 βˆ’a > βˆ’b (c) 0 < 1 (d) If a > 0, then aβˆ’1 > 0 (e) If a < 0, then aβˆ’1 < 0 (f) If a < b and a and b are both positive, then aβˆ’1 > bβˆ’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 β‰₯ 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 β‰₯ 0 (b) a2 = 0 iff a = 0 (c) a2 > 0 iff a > 0 (d) (βˆ’a)2 = a2 (e) (aβˆ’1)2 = 1/a2 (f) If a2 = b2, then a = Β±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 √a2 = 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| β‰₯ 0.

    • b) |a| = 0 iff a = 0.

    • c) |a| > 0 iff a 6= 0.

    • d) | βˆ’ a| = |a|.

    • e) |a| = √a2.

    • f) |a| = max{a, βˆ’a}.

    • g) |aβˆ’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 = Β±b.

    • k) If a and b are both nonnegative, then |a| β‰₯ |b| iff a β‰₯ b.

    • l) If a and b are both negative, then |a| β‰₯ |b| iff a ≀ b.

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

    • n) (The reverse triangle inequality) |a| βˆ’ |b| ≀ |a βˆ’ 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 β‰₯ 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

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