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d) The product of two numbers, a and b is zero if and only if at least one of them is zero; e) For any given a, beRt satisfying a < b, one has a? < b. [Recall that the symbol > is defined as follows: for a, y E R, r > y means r-y E R*];

d) The product of two numbers, a and b is zero if and only if at least one of them is zero; e) For any given a, beRt satisfying a < b, one has a? < b. [Recall that the symbol > is defined as follows: for a, y E R, r > y means r-y E R*];

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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part D E F

1. Using exclusively the axiomatic description of real numbers enclosed (see at the end of this document), derive the following properties: a) The multiplication ab produces zero if and only if at least one of the elements a or b of the product is zero. b) The product of two negative elements produces a positive element, while a negative times a positive yields a negative. c) For any given a e R, there is a unique opposite associated to a; show also that for any given 0 a € R, there is a unique inverse associated to a. (Note that these uniqueness results justify the notation opposite of a, and ! for the (unique) inverse of a (for a + 0)); -a for the (unique) d) The product of two numbers, a and b is zero if and only if at least one of them is zero; e) For any given a, bERt satisfying a < b, one has a' < b². [Recall that the symbol > is defined as follows: for I, y ER, r > y means r – y ER+); f) Explain the scope of the requirement 1 +0 contained in Axiom (iv) of the axiomatic description of real numbers.
Transcribed Image Text:1. Using exclusively the axiomatic description of real numbers enclosed (see at the end of this document), derive the following properties: a) The multiplication ab produces zero if and only if at least one of the elements a or b of the product is zero. b) The product of two negative elements produces a positive element, while a negative times a positive yields a negative. c) For any given a e R, there is a unique opposite associated to a; show also that for any given 0 a € R, there is a unique inverse associated to a. (Note that these uniqueness results justify the notation opposite of a, and ! for the (unique) inverse of a (for a + 0)); -a for the (unique) d) The product of two numbers, a and b is zero if and only if at least one of them is zero; e) For any given a, bERt satisfying a < b, one has a' < b². [Recall that the symbol > is defined as follows: for I, y ER, r > y means r – y ER+); f) Explain the scope of the requirement 1 +0 contained in Axiom (iv) of the axiomatic description of real numbers.
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