Unit 2: Using Algebra and Graphs to Describe Relationships
... • The domain and range of a function may be continuous or discrete depending on the situation. – If a variable can take all values within some interval it is called a continuous variable. • For example, distance traveled as a function of time is a continuous function since there would be no gaps in ...
... • The domain and range of a function may be continuous or discrete depending on the situation. – If a variable can take all values within some interval it is called a continuous variable. • For example, distance traveled as a function of time is a continuous function since there would be no gaps in ...
The Equal Sign, Equations, and Functions, Draft 3 by Solomon
... assertion that two mathematical objects are actually the same; it may be either true or false. An equation involving sets is true if the sets on the two sides of the equal sign have exactly the same elements, and otherwise it is false. With the notation as above, if S1 = {1, 3} and T1 = {2, 4}, then ...
... assertion that two mathematical objects are actually the same; it may be either true or false. An equation involving sets is true if the sets on the two sides of the equal sign have exactly the same elements, and otherwise it is false. With the notation as above, if S1 = {1, 3} and T1 = {2, 4}, then ...
Manipulating inequalities
... Numbers can be represented on a number line. If a < b then equivalently, b > a. The symbol > means ‘greater than’; for example, since 6 is greater than 4 we can write 6 > 4. Given any number, all numbers to the right of it on the line are greater than the given number. The symbol < means ‘less than’ ...
... Numbers can be represented on a number line. If a < b then equivalently, b > a. The symbol > means ‘greater than’; for example, since 6 is greater than 4 we can write 6 > 4. Given any number, all numbers to the right of it on the line are greater than the given number. The symbol < means ‘less than’ ...
Manipulating inequalities
... Numbers can be represented on a number line. If a < b then equivalently, b > a. The symbol > means ‘greater than’; for example, since 6 is greater than 4 we can write 6 > 4. Given any number, all numbers to the right of it on the line are greater than the given number. The symbol < means ‘less than’ ...
... Numbers can be represented on a number line. If a < b then equivalently, b > a. The symbol > means ‘greater than’; for example, since 6 is greater than 4 we can write 6 > 4. Given any number, all numbers to the right of it on the line are greater than the given number. The symbol < means ‘less than’ ...
Completed versus Incomplete Infinity in Arithmetic
... infinity, so if we want to understand infinity we should try to understand the numbers. Rather than use the symbols 1, 2, etc. originating in India, let us use the notation of mathematical logic: the numbers are 0, S0, SS0, SSS0, . . . where S can be read as “successor”. This notation expresses clea ...
... infinity, so if we want to understand infinity we should try to understand the numbers. Rather than use the symbols 1, 2, etc. originating in India, let us use the notation of mathematical logic: the numbers are 0, S0, SS0, SSS0, . . . where S can be read as “successor”. This notation expresses clea ...
Cardinality, countable and uncountable sets
... looks very much like “an equivalence relation in the class of all sets”, and indeed this can be formalized in axiomatic set theory, but we’ll leave that for the advanced course. The notion of “cardinality” of a set was develop in the late 19th/early 20th centuries by the German mathematician Georg C ...
... looks very much like “an equivalence relation in the class of all sets”, and indeed this can be formalized in axiomatic set theory, but we’ll leave that for the advanced course. The notion of “cardinality” of a set was develop in the late 19th/early 20th centuries by the German mathematician Georg C ...
A constructive approach to nonstandard analysis*
... paper, we could say that they work with the reduced power of the reals, modulo the cofinite filter on the natural numbers, whereas Robinson’s approach amounts to using instead a nonprincipal ultrafilter. Thus in the former approach two sequences of real numbers are identified if they eventually agre ...
... paper, we could say that they work with the reduced power of the reals, modulo the cofinite filter on the natural numbers, whereas Robinson’s approach amounts to using instead a nonprincipal ultrafilter. Thus in the former approach two sequences of real numbers are identified if they eventually agre ...
CARLOS AUGUSTO DI PRISCO The notion of infinite appears in
... The construction of the model M [g] and the proof that it has the desired properties is quite elaborate. Certain elements of the model M are used as “names” for elements of M [g]. Which set is the object named by a name τ depends on the generic g, and given g, the model M [g] is the collection of se ...
... The construction of the model M [g] and the proof that it has the desired properties is quite elaborate. Certain elements of the model M are used as “names” for elements of M [g]. Which set is the object named by a name τ depends on the generic g, and given g, the model M [g] is the collection of se ...
Chapter 2 Propositional Logic
... that the truth table for ¬(p ∧ q) and ¬p ∨ ¬q are the same, which will say that both compound propositions are equivalent. Let us compute in details the first row of the truth table for ¬(p ∧ q) and ¬p ∨ ¬q: if both p and q are true, then ¬p and ¬q are false, and p ∧ q is true, since it is true exac ...
... that the truth table for ¬(p ∧ q) and ¬p ∨ ¬q are the same, which will say that both compound propositions are equivalent. Let us compute in details the first row of the truth table for ¬(p ∧ q) and ¬p ∨ ¬q: if both p and q are true, then ¬p and ¬q are false, and p ∧ q is true, since it is true exac ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.