Exam 1 Flashcards
Three of the most important kinds of sentences in mathematics are _______ statements, _______ statements, and _______ statements.
Universal
Conditional
Existential
Says that a certain property is true for all elements in a set.
Universal statement
Says that if one thing is true then some other thing also has to be true.
Conditional statement
Given a property that may of may not be true, an _________________ says that there is at least one thing for which the property is true.
Existential statement
Contain some variation of the words “for all” and conditional statements contain versions of the words “if-then”.
Universal statements
A _____________________ is a statement that is both universal and conditional. Here is an example:
For all animals a, If a is a dog, then a is a mammal.
Universal conditional statement
One of the most important facts about universal conditional statements is that they can be rewritten in ways that make them appear to be purely ________ or purely _________.
Universal
Conditional
A statement that is universal because it’s first part says that a certain property is true for all objects of a given type, and it is existential because it’s second part asserts the existence of something. For example:
Every real number has and additive inverse.
Universal existential statement
“______________________” asserts the existence of something- and additive inverse- for each real number
Has an additive inverse
A statement that is existential because it’s first part asserts that a certain object exists and is universal because it’s second part says that the object satisfies a certain property for all things of a certain kind.
Existential universal statement
Example:
There is a positive integer that is less than or equal to every positive integer
What statement does this true example refer too?
Existential universal statement
Some of the most important mathematical concepts, such as the definition limit of a sequence, can only be defined using phrases that are ________, ________, and __________, and they require the use of all three phases “______”, “_______”, and “_______”.
Universal
Existential
Conditional
For all
There is
If-then
A set may be specified using the _____________ by writing all the elements between braces. Example:
{1,2,3}
Set-roster notation
An infinite set symbol. Example:
{1,2,3,…}
Ellipsis
__ set of all real numbers
— set of all integers
— set of all rational numbers, or quotients of integers
R
Z
Q
The number 0 to a middle point, called the _______.
Origin
A unit of distance is marked off, and each point to the right of the origin corresponds to a ________________ found by the computing its distance from the origin.
Positive real number
Each point to the left of the origin corresponds to a _______________, which is denoted by computing its distance from the origin and putting a minus sign in front of the resulting number.
Negative real number
The set of real numbers is therefore divided into three parts: set of _________ numbers, set of ________ numbers, and the number __, which is neither positive or negative.
Positive real
Negative real
0
The name _____________ comes from the distinction between continuous and discrete mathematical objects.
Discrete mathematics
__ denotes the set of all real numbers
__ denotes the set of all integers
__ denotes the set of all positive integers
R
Z
Z+
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 A.
Proper subset
Given sets A and B, the _____________ of A and B, denoted A x B and read “A cross B”.
Cartesian product
The term ___________ is often used to refer to a plane with this coordinate system.
Cartesian plane
