MMW chapter2

Mathematics

spoken and written natural languages for expressing mathematical language

Mathematical language

system used to communicate mathematical ideas.

precise and concise/poor understanding of the language.

Mathematical language is

Mathematical notation

used for formulas has its own grammar and shared by mathematicians anywhere in the globe.

expression or mathematical expression

finite combination of symbols that is well-defined according to rules that depend on the context.

Symbols

can designate numbers, variables, operations, functions, brackets, punctuations, and groupings to help determine order of operations, and other aspects of mathematical syntax.

Expression

correct arrangement of mathematical symbols to represent the object of interest, does not contain a complete thought, and cannot be determined if it is true or false.

numbers, sets, and functions.

Some types of expressions are

sentence or mathematical sentence

a statement about two expressions, either using numbers, variables, or a combination of both.

Mathematical Convention

fact, name, notation, or usage which is generally agreed upon by mathematicians.

Set theory

branch of mathematics that studies sets or the mathematical science of the infinite.

George Cantor

considered as the founder of set theory as a mathematical discipline.

set

well-defined collection of objects.

elements or members

The objects are called the _________ of the set.

Roster Method or tabulation method

elements of the set are enumerated and separated by a comma

Rule Method or set builder

descriptive phrase is used to describe the elements or members of the set

Finite set

set whose elements are limited or countable, and the last element can be identified.

Infinite set

set whose elements are unlimited or uncountable, and the last element cannot be specified.

unit set or singleton.

set with only one element

empty set or null set

unique set with no elements, it is denoted by the symbol Æ or { }.

Universal set

all sets under investigation in any application of set theory are assumed to be contained in some large fixed set, denoted by the symbol U.

cardinal number

of a set is the number of elements or members in the set, the cardinality of set A is denoted by n(A)

Venn Diagram

a pictorial presentation of relation and operations on set.

set diagrams

venn diagram is also called

Venn Diagram

it show all hypothetically possible logical relations between finite collections of sets

John Venn

Venn Diagram is Introduced by

subset

If A and B are sets, A is called ____ of B, if and only if, every element of A is also an element of B.

proper subset

Let A and B be sets. A is a _________ of B, if and only if, every element of A is in B but there is at least one element of B that is not in A.

equals

Given set A and B, A _____ B, written, if and only if, every element of A is in B and every element of B is in A.

power set

Given a set S from universe U, the_____ of S denoted by Ã(S), is the collection (or sets) of all subsets of S.

Theorem 1.2

A Set with No Elements is a Subset of Every Set: If  Æ is a set with no   elements and A is any set, then Æ Í A.

Theorem 1.3

For all sets A and B, if A Í B then Ã(A) Í Ã(B).

Theorem 1.4

Power Sets: For all integers n, if a set S has n elements then Ã(S) has 2ⁿ elements.

union

The ____ of A and B, denoted AÈB, is the set of all elements x in U such that x is in A or x is in B.

intersection

The _____ of A and B, denoted AÇB, is the set of all elements x in U such that x is in A and x is in B.

complement

The _____ of A (or absolute complement of A), denoted A’, is the set of all elements x in U such that x is not in A.

difference or relative complement

The _____ of A and B (or _______ of B with respect to A), denoted A ~ B, is the set of all elements x in U such that x is in A and x is not in B.

symmetric difference

If set A and B are two sets, their ________ as the set consisting of all elements that belong to A or to B, but not to both A and B.

disjoint

Two set are called _____ (or non-intersecting) if and only if, they have no elements in common.

ordered pair

In the ______ (a, b), a is called the first component and b is called the second component. In general, (a, b) ¹ (b, a).

relation

A ___ is a set of ordered pairs.

domain

of R is the set dom R

  dom R = {a Î A| (a, b) Î R for some b Î B}.

image

(or range) of R

  im R = {b Î B| (a, b) Î R for some a Î A}.

Function

special kind of relation helps visualize relationships in terms of graphs and make it easier to interpret different behavior of variables

function

A ______ is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.

domain

The set X is called the _____ of the function.

value

For each element of x in X, the corresponding element y in Y is called the _______ of the function at x, or the image of x.

Range

set of all images of the elements of the domain is called the of the function. A function can map from one set to another.

group

A _______ is a set of elements, with one operation, that satisfies

group

ordered pair (G, é) where G is a set and é is a binary operation on G satisfying the four properties. 

Closure property

If any two elements are combined using   the operation, the result must be an element of the set.   a é b = c Î G, for all a, b, c Î G.

Associative property

(a é b) é c = a é (b é c), for all a, b,   c Î G.

Identity property

There exists an element e in G, such that   for all a Î G, a é e = e é a.

Inverse property

For each a Î G there is an element a^–1 of   G, such that a é a^–1 = a^–1 é a = e.

Mathematical logic or symbolic logic

branch of mathematics with close connections to computer science.

Set Theory/Proof Theory/Recursion Theory/Model Theory

Four Divisions:

Aristotle

generally regarded as the Father of Logic

statement or proposition

declarative sentence which is either true or false, but not both.

truth value

The ____ of the statements is the truth and falsity of the statement.

formal propositional

A _______ written using propositional logic notation, p, q, and r are used to represent statements.

Logical connectives

used to combine simple statements which are referred as compound statements.

compound statement

statement composed of two or more simple statements connected by logical connectives

conjunction

statement p and q is the compound statement “p and q.”

disjunction

statement p, q is the compound statement “p or q.”

negation

statement p is denoted by p, where  is the symbol for “not.”

conditional or implication)

statement p and q is the compound statement “if p then q.”

biconditional

statement p and q is the compound statement “p if and only if q.”

exclusive-or

statement p and q is the compound statement “p exclusive or q.”

predicate or open statements

a statement whose truth depends on the value of one or more variables.

universe of discourse

Predicates become propositions once every variable is bound by assigning a  ___________

function-like notation

A predicate can also be denoted by a

propositional function

sentence P(x); it becomes a statement only when variable x is given particular value.

universe of discourse

The _____________ for the variable x is the set of positive real numbers for the proposition

bound

Binding variable is used on the variable x, we can say that the occurrence of this variable is

scope

The _____ of a quantifier is the part of an assertion in which variables are bound by the quantifier.

universal quantifier

The symbol (for all) is called the

existential quantifier

The symbol (there exists) is called the

free

A variable is said to be ___, if an occurrence of a variable is not bound.