By Walter K. Buhler
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Extra resources for Gauss: a biographical study
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In particular this phrase is used along with an existence quantifier as “such that” is another way to say “which has the property that” When we write a statement using “for every” or “there exists,” the first of these denotes that a property is going to follow and that this property holds for all. The second exhorts that there is a time when the following property holds. ” Note that if in this latter sentence we omitted the “such that” and simply wrote “There exists a real number x, x2 is non-negative” our sentence would not be grammatically correct.
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1. Terminology and Goals 29 prove it. Addition is simply defined to have the commutative property and so this is considered to be an axiom. In defining any number system (such as using addition with the real numbers) you have to set up some basic rules (axioms) and definitions and then build upon those by proofs. In this class we will take some properties as axioms and we will try to clearly state those so that you are well aware what properties must be proven, and what properties we are assuming to be true.
Disprove the following statement: If x and y are nonnegative integers, √ √ √ then x + y = x + y. 8. Disprove the following statement: If x is irrational, then x2 is irrational. 9. Provide a counterexample to show that the following statement is not true: Let a ∈ Z. If a2 ≡ 4 mod 5, then a ≡ 2 mod 5. 10. Disprove the following statement concerning integers a, b, and c: If a|bc, then a|c. 11. Disprove the following statement: If x, y ∈ Z, then 3x + 2y = 1. 12. Let a, b, c ∈ Z. Disprove the following statement: If a < b, then ac < bc.