Completeness of the real numbers
... above, let L = sup({xn ; n ≥ 0}). It is easy to show that xn → L. The converse is true! Theorem. Let F be an odered field with the monotone convergence property. Then if A ⊂ F is non-empty and bounded above, there exists L ∈ F so that L = sup(A). Proof. (i) Let a1 ∈ A be arbitrary, M1 ∈ F an upper b ...
... above, let L = sup({xn ; n ≥ 0}). It is easy to show that xn → L. The converse is true! Theorem. Let F be an odered field with the monotone convergence property. Then if A ⊂ F is non-empty and bounded above, there exists L ∈ F so that L = sup(A). Proof. (i) Let a1 ∈ A be arbitrary, M1 ∈ F an upper b ...
Lecture One: Overview and Fundamental Concepts
... Disproof by Counterexample A counterexample to x P(x) is an object c so that P(c) is false. Statements such as x (P(x) Q(x)) can be disproved by simply providing a counterexample. ...
... Disproof by Counterexample A counterexample to x P(x) is an object c so that P(c) is false. Statements such as x (P(x) Q(x)) can be disproved by simply providing a counterexample. ...
Document
... positive integers. The second row contains all the fractions with denominator equal to 2. The third row contains all the fractions with denominator equal to 3, etc. ...
... positive integers. The second row contains all the fractions with denominator equal to 2. The third row contains all the fractions with denominator equal to 3, etc. ...
8 Addition and Subtraction of Whole Numbers
... where A − B = {x ∈ A and x 6∈ B}. •The Missing-Addend Model This model relates subtraction and addition. In this model, given two whole numbers a and b we would like to find the whole number c such that c + b = a. We call c the missing-addend and its value is c = a − b. Cashiers often use this model ...
... where A − B = {x ∈ A and x 6∈ B}. •The Missing-Addend Model This model relates subtraction and addition. In this model, given two whole numbers a and b we would like to find the whole number c such that c + b = a. We call c the missing-addend and its value is c = a − b. Cashiers often use this model ...
Note 7 - Counting Techniques
... Two numbers that add up to 9 are placed in sets as follows: A1 = {1, 8}, ...
... Two numbers that add up to 9 are placed in sets as follows: A1 = {1, 8}, ...
MAT 1348/1748 SUPPLEMENTAL EXERCISES 1 Propositional Logic
... 5. Show that at least one of the real numbers a1 , a2 , ..., an is greater than or equal to the average of these numbers. 6. Using a proof by contradiction, show that between any two distinct rational numbers there are infinitely many rational numbers. ...
... 5. Show that at least one of the real numbers a1 , a2 , ..., an is greater than or equal to the average of these numbers. 6. Using a proof by contradiction, show that between any two distinct rational numbers there are infinitely many rational numbers. ...
Cantor - Muskingum University
... Cantor began to work on the Continuum Hypothesis, but was not able to make much progress. The Continuum Hypothesis was a theory that stated that the cardinality of the real numbers was next in order after the natural numbers. The inability to resolve this worsened his mental state. It was improved ...
... Cantor began to work on the Continuum Hypothesis, but was not able to make much progress. The Continuum Hypothesis was a theory that stated that the cardinality of the real numbers was next in order after the natural numbers. The inability to resolve this worsened his mental state. It was improved ...
Chapter 2: Boolean Algebra and Logic Gates
... • Comparing Boolean algebra with arithmetic and ordinary algebra 1. Huntington postulates do not include the associative law. However, this law holds for Boolean algebra and can be derived (for both operators) from the other postulates. 2 The distributive law of + over • (i.e., ...
... • Comparing Boolean algebra with arithmetic and ordinary algebra 1. Huntington postulates do not include the associative law. However, this law holds for Boolean algebra and can be derived (for both operators) from the other postulates. 2 The distributive law of + over • (i.e., ...