Exam 1 Flashcards

1
Q

Three of the most important kinds of sentences in mathematics are _______ statements, _______ statements, and _______ statements.

A

Universal
Conditional
Existential

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Says that a certain property is true for all elements in a set.

A

Universal statement

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Says that if one thing is true then some other thing also has to be true.

A

Conditional statement

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

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.

A

Existential statement

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Contain some variation of the words “for all” and conditional statements contain versions of the words “if-then”.

A

Universal statements

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

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.

A

Universal conditional statement

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

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 _________.

A

Universal

Conditional

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

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.

A

Universal existential statement

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

“______________________” asserts the existence of something- and additive inverse- for each real number

A

Has an additive inverse

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

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.

A

Existential universal statement

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Example:

There is a positive integer that is less than or equal to every positive integer

What statement does this true example refer too?

A

Existential universal statement

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

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 “_______”.

A

Universal
Existential
Conditional

For all
There is
If-then

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

A set may be specified using the _____________ by writing all the elements between braces. Example:

{1,2,3}

A

Set-roster notation

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

An infinite set symbol. Example:

{1,2,3,…}

A

Ellipsis

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

__ set of all real numbers
— set of all integers
— set of all rational numbers, or quotients of integers

A

R
Z
Q

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

The number 0 to a middle point, called the _______.

A

Origin

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

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.

A

Positive real number

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

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.

A

Negative real number

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

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.

A

Positive real
Negative real
0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

The name _____________ comes from the distinction between continuous and discrete mathematical objects.

A

Discrete mathematics

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

__ denotes the set of all real numbers
__ denotes the set of all integers
__ denotes the set of all positive integers

A

R
Z
Z+

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

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.

A

Proper subset

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Given sets A and B, the _____________ of A and B, denoted A x B and read “A cross B”.

A

Cartesian product

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

The term ___________ is often used to refer to a plane with this coordinate system.

A

Cartesian plane

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
And _________ is a sequence of statements.
Argument
26
A ________ is a sequence of statement forms.
Argument form
27
When an argument is valid and it’s premises are true, the truth of the conclusion is said to be _________ or _________.
Inferred | Deduced
28
If a conclusion “ain’t necessarily so”, then is is it a _____________.
Valid deduction
29
An argument form consisting of two premises and a conclusion is called a ____________
Syllogism
30
The first and second premises are called the ______________ and ______________.
Major premise | Minor premise
31
The most famous form of syllogism in logic is called ______________. Is has the following form: If p then a. p q
Modus ponens
32
A valid argument form called _____________. It has the following: If p then q. ~q ~p
Modus tollens
33
A form of argument that is valid. This modus ponens and modus tollens are both rules of inference.
Rule of inference
34
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
35
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
36
___________ of q by p is “if p then q” and is denoted p->q
Conditional
37
p->q What is p of the conditional
Hypothesis
38
p->q What is q of the conditional
Conclusion or consequent
39
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
40
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
41
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
42
By definition, p->q is false if, and only if, it’s hypothesis,p, is ______ and it’s conclusion,q, is ______.
True | False
43
The negation of if p then q is logically equivalent to ___________
p and not q
44
The __________________ of p->q is ~q->~p
Contrapositive
45
A conditional statement is logically equivalent to its ?
Contrapositive
46
A conditional statement and it’s converse are ?
Not logically equivalent
47
A conditional statement and it’s inverse are ?
Not logically equivalent
48
A converse and the inverse of a conditional statement are ?
Logically equivalent to each other
49
Given statement variables p and q, the _____________ of p and q is “p If, and only if,q” and is denoted pq
Biconditional
50
The addriviation iff means?
If and only if
51
is coequal with ->. The only way to indicate precedence between them is to use ____________
Parentheses
52
What is the order of operations for logical operators
1. ~ 2. V, ^ 3. ->,
53
r is a ______________ condition for s Means “if r then s”
Sufficient
54
r is a ______________ condition for s Means “if not r then not s”
Necessary
55
“r is a sufficient condition for s” means that the occurrence of r is _________ to guarantee the occurrence of s
Sufficient
56
A sentence that is true or false but not both
Statement
56
~p is read not p and is called
The negation of p
56
p ^ q is read p and q and is called
The conjunction of p and q
56
p v q is read p or q and is called
Disjunction of p and q
57
Statement form displays the truth values
Truth table
58
Two statements forms are called __________ If and only if they have identical truth values
Logically equivalent
59
Symbol - - - Means?
Logically equivalent
60
Two logical equivalences are known as
De Morgan’s laws
61
~(p^q) logically equivalent to ~p v ~q ~(p v q) logically equivalent to ~p ^ ~q
De Morgan’s law
62
A tautology is a statement that is always _____
True
63
A contradiction is a statement form that is always _________
False
64
A row of the truth table in which all the premises are true is called a
Critical row
65
The operation of a black box is completely specified by constructing an
Input/output table
66
An efficient method for designing more complicated circuits is to build them by connecting less complicated _________ circuits
Black box
67
A circuit with one input signs and one output signal Looks like |>•
NOT-gate
68
A circuit with two input signals and one output signal Looks like a D
AND-gate
69
A circuit with two input signals and one output signal Looks like |>
OR-gate
70
Gates can be combined into circuits in a variety of ways called
Combination circuit
71
Expression composed of ~,^, and V is called a
Boolean expression
72
p -> q ~p -> ~q
Inverse
73
p -> q q -> p
Converse