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
And _________ is a sequence of statements.
Argument
A ________ is a sequence of statement forms.
Argument form
When an argument is valid and it’s premises are true, the truth of the conclusion is said to be _________ or _________.
Inferred
Deduced
If a conclusion “ain’t necessarily so”, then is is it a _____________.
Valid deduction
An argument form consisting of two premises and a conclusion is called a ____________
Syllogism
The first and second premises are called the ______________ and ______________.
Major premise
Minor premise
The most famous form of syllogism in logic is called ______________. Is has the following form:
If p then a.
p
q
Modus ponens
A valid argument form called _____________. It has the following:
If p then q.
~q
~p
Modus tollens
A form of argument that is valid. This modus ponens and modus tollens are both rules of inference.
Rule of inference
Let p and q be statements. A sentence of the form “if p then q” is denoted symbolically by “p->q”; p is called the ___________ and q is called the ____________.
Hypothesis
Conclusion
If 10 is divisible by 10,
|
Hypothesis
then 10 is divisible by 5
|
Conclusion
Such a sentence is called ___________ because the truth of statement q is conditioned on the truth of statement p.
Conditional
___________ of q by p is “if p then q” and is denoted p->q
Conditional
p->q
What is p of the conditional
Hypothesis
p->q
What is q of the conditional
Conclusion or consequent
A conditional statement that is true by virtue of the fact that it’s hypothesis is false is often called?
Vacuously true
Or
True by default
Thus the statement “If you show up for work Monday morning, then you will get the job” is vacuously true if you ________________ for work Monday morning.
Do not show up
In general, when the “if” part of an if-then statement is false, the statement as a whole is said to be ______, regardless of whether the conclusion is true or false.
True
By definition, p->q is false if, and only if, it’s hypothesis,p, is ______ and it’s conclusion,q, is ______.
True
False
The negation of if p then q is logically equivalent to ___________
p and not q
The __________________ of p->q is ~q->~p
Contrapositive
A conditional statement is logically equivalent to its ?
Contrapositive
A conditional statement and it’s converse are ?
Not logically equivalent
A conditional statement and it’s inverse are ?
Not logically equivalent
A converse and the inverse of a conditional statement are ?
Logically equivalent to each other
Given statement variables p and q, the _____________ of p and q is “p If, and only if,q” and is denoted pq
Biconditional
The addriviation iff means?
If and only if
is coequal with ->. The only way to indicate precedence between them is to use ____________
Parentheses
What is the order of operations for logical operators
- ~
- V, ^
- ->,
r is a ______________ condition for s Means “if r then s”
Sufficient
r is a ______________ condition for s Means “if not r then not s”
Necessary
“r is a sufficient condition for s” means that the occurrence of r is _________ to guarantee the occurrence of s
Sufficient
A sentence that is true or false but not both
Statement
~p is read not p and is called
The negation of p
p ^ q is read p and q and is called
The conjunction of p and q
p v q is read p or q and is called
Disjunction of p and q
Statement form displays the truth values
Truth table
Two statements forms are called __________ If and only if they have identical truth values
Logically equivalent
Symbol -
-
-
Means?
Logically equivalent
Two logical equivalences are known as
De Morgan’s laws
~(p^q) logically equivalent to ~p v ~q
~(p v q) logically equivalent to ~p ^ ~q
De Morgan’s law
A tautology is a statement that is always _____
True
A contradiction is a statement form that is always _________
False
A row of the truth table in which all the premises are true is called a
Critical row
The operation of a black box is completely specified by constructing an
Input/output table
An efficient method for designing more complicated circuits is to build them by connecting less complicated _________ circuits
Black box
A circuit with one input signs and one output signal
Looks like |>•
NOT-gate
A circuit with two input signals and one output signal
Looks like a D
AND-gate
A circuit with two input signals and one output signal
Looks like |>
OR-gate
Gates can be combined into circuits in a variety of ways called
Combination circuit
Expression composed of ~,^, and V is called a
Boolean expression
p -> q
~p -> ~q
Inverse
p -> q
q -> p
Converse
