The Natural Number System: Induction and Counting
... Claim 2. [0, ∞) is inductive. Let x ∈ [0, ∞). Then, by definition of the interval [0, ∞), x ≥ 0. By Axiom 6 of the real number system (and the defining property of 0), x + 1 ≥ 1. We proved earlier that 1 > 0. So by transitivity, x + 1 > 0. Thus, we have shown that [0, ∞) is inductive: if x ∈ [0, ∞), ...
... Claim 2. [0, ∞) is inductive. Let x ∈ [0, ∞). Then, by definition of the interval [0, ∞), x ≥ 0. By Axiom 6 of the real number system (and the defining property of 0), x + 1 ≥ 1. We proved earlier that 1 > 0. So by transitivity, x + 1 > 0. Thus, we have shown that [0, ∞) is inductive: if x ∈ [0, ∞), ...
Math 90 Lecture Notes Chapter 1
... B. The law of opposites: Any pair of opposite numbers, additive inverses, always adds up to what number? _________________ C. Let’s think about a new way of thinking about subtraction. Some simple examples will help us here: 1. You have $700 in your checking account and write a check for $200. How m ...
... B. The law of opposites: Any pair of opposite numbers, additive inverses, always adds up to what number? _________________ C. Let’s think about a new way of thinking about subtraction. Some simple examples will help us here: 1. You have $700 in your checking account and write a check for $200. How m ...
Document
... Proof: Check exhaustively and find that 211 is prime. QED. Example 2. Each of the numbers 288, 198, and 387 is divisible by 9. Proof: Check that 9 divides each of the numbers. QED. Conditional Proof Most statements we prove are conditionals. We start by assuming the antecedent is true. Then we try t ...
... Proof: Check exhaustively and find that 211 is prime. QED. Example 2. Each of the numbers 288, 198, and 387 is divisible by 9. Proof: Check that 9 divides each of the numbers. QED. Conditional Proof Most statements we prove are conditionals. We start by assuming the antecedent is true. Then we try t ...
Section.1.1
... Proof: Check exhaustively and find that 211 is prime. QED. Example 2. Each of the numbers 288, 198, and 387 is divisible by 9. Proof: Check that 9 divides each of the numbers. QED. Conditional Proof Most statements we prove are conditionals. We start by assuming the antecedent is true. Then we try t ...
... Proof: Check exhaustively and find that 211 is prime. QED. Example 2. Each of the numbers 288, 198, and 387 is divisible by 9. Proof: Check that 9 divides each of the numbers. QED. Conditional Proof Most statements we prove are conditionals. We start by assuming the antecedent is true. Then we try t ...
Fundamental units and consecutive squarefull numbers,
... Ankeny–Artin–Chowla conjecture and the Mordell conjecture are true for almost all primes p (in the sense of density). More precisely, we show that the set of primes p ≤ x for which the conjecture is false is O(xθ ) for some θ < 1. Our methods are elementary and do not use anything deeper than the th ...
... Ankeny–Artin–Chowla conjecture and the Mordell conjecture are true for almost all primes p (in the sense of density). More precisely, we show that the set of primes p ≤ x for which the conjecture is false is O(xθ ) for some θ < 1. Our methods are elementary and do not use anything deeper than the th ...
presentation source
... Random Sequences • If we choose a, c and m carefully then all numbers in the range from 0 to m-1 will appear in the sequence • to have the numbers in the range 0 r <1, the generator returns xm/m which is always < 1 • there is no necessary and sufficient test for the randomness of a finite sequenc ...
... Random Sequences • If we choose a, c and m carefully then all numbers in the range from 0 to m-1 will appear in the sequence • to have the numbers in the range 0 r <1, the generator returns xm/m which is always < 1 • there is no necessary and sufficient test for the randomness of a finite sequenc ...
Full text
... It is clear from (2.26) that A(n9X) is the number of X1 partitions of Z n . It is also clear from (2.7) and the above comments that b{n9 k + 1, X) is the number of Xx partitions of Zn into k blocks such that at least two elements of Zn are placed in the open boxes. Definition (2.4) furnishes another ...
... It is clear from (2.26) that A(n9X) is the number of X1 partitions of Z n . It is also clear from (2.7) and the above comments that b{n9 k + 1, X) is the number of Xx partitions of Zn into k blocks such that at least two elements of Zn are placed in the open boxes. Definition (2.4) furnishes another ...
2-1 - SPX.org
... An Irrational Number is a number that cannot be expressed in the form a/b, where a and b are integers and b 0. (Decimals that do not repeat or do not terminate.) The set of rational numbers and irrational numbers together form the set of Real Numbers. Recall from 2-1: ...
... An Irrational Number is a number that cannot be expressed in the form a/b, where a and b are integers and b 0. (Decimals that do not repeat or do not terminate.) The set of rational numbers and irrational numbers together form the set of Real Numbers. Recall from 2-1: ...