Basic Properties of Real Numbers: Exercise Supplement
In this section, let’s turn our attention to proofs of basic algebraic and ordering properties of real numbers taken from book entitled ” Mathematical Writing and Thinking” by Randall Maddox. We might want to take a look at assumsions A1-A18 below to show some proofs of statements related with real numbers. My friend (Sri Rejeki) and I has tried to carry out the exercise about real number given in that book, and if you don’t mind, please evaluate our works by leaving your comments. You can download by pressing “download” button after the description of the assumptions below. Thank You,.
These are the assumptions (A1 – A18):(A1) Properties of equality; (a) For every (Reflexive property); (b) If a = b, then b = a (Symmetric property); (c) If a = b and b = c, then a = c (Transitive property). (A2) Addition is well defined; That is, if a,b,c,d elements of R, where a=b, and c=d, then a+c=b+d (A3) Closure property of addition; For every a,b elements of R, a+b element of R (A4) Associative property of addition; For every a,b,c elements of R, (a+b)+c=a+(b+c) (A5) Commutative property of addition; For every a,b elements of R, a+b=b+a (A6) Existence of an additive identity; There exists an element 0 ∈ R with the property that a + 0 = a for every a ∈ R. (A7) Existence of additive inverses; For every a ∈ R, there exists some b ∈ R such that a + b = 0. Such an element b is called an additive inverse of a, and is typically denoted −a to show its relationship to a. We do not assume that only one such b exists. (A8) Multiplication is well defined; That is, if a, b, c, d ∈ R, where a = b and c = d, then ac = bd. (A9) Closure property of multiplication; For all a, b ∈ R , a∙b ∈ R. The closure property of multiplication also holds for N, W, Z, and Q. (A10) Associative property of multiplication; For every a, b, c ∈ R, (a·b)·c = a·(b·c) or (a∙b)∙c = a∙(b∙c). (A11) Commutative property of multiplication; For every a, b ∈ R, a·b = b·a. (A12) Existence of a multiplicative identity; here exists an element 1 ∈ R with the property that a·1 = a for every a ∈ R. (A13) Existence of multiplicative inverses; For every a ∈ R except a = 0, there exists some b ∈ R such that a·b = 1. Such an element b is called a multiplicative inverse of a and is typically denoted a-1 to show its relationship to a. As with additive inverses, we do not assume that only one such b exists. Furthermore, the assumption that a-1 exists for all a ≠ 0 does not assume that zero does not have a multiplicative inverse. It says nothing about zero at all. (A14) Distributive property of multiplication over addition; For every a, b, c ∈ R, a (b + c) = (a∙b) + (a∙c) = a∙b + a∙c, where the multiplication is assumed to be done before addition in the absence of parentheses. (A15) 1 ≠ 0 (A16) Trichotomy law; For any a ∈ R, exactly one of the following is true: (a) a > 0, in which case we say a is positive; (b) a = 0; (c) 0 > a, in which case we say a is negative. (A17) If a > 0 and b > 0, then a + b > 0. That is, the set of positive real numbers is closed under addition. (A18) If a > 0 and b > 0, then a∙b > 0. That is, the set of positive real numbers is closed under multiplication