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Mathematics in the Modern World > Module 5. Mathematical Logic > Flashcards

Module 5. Mathematical Logic Flashcards

1
Q

deals with the scientific method of judging the truth or falsity of statements

A

Logic

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2
Q

body of statements or mathematical sentences

A

Mathematics

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3
Q

important in understanding mathematics as a logical system

A

symbolic language

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4
Q

basis of all mathematic reasoning

A

logic

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5
Q

one of the first mathematicians to make a serious study of symbolic logic

A

Gottfried Wilhelm Leibniz (1646-1716)

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6
Q

Leibniz tried to advance the study of logic from a merely __ subject to a formal __ subject

A
  • philosophical
  • mathematical
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7
Q

Was Leibniz able to achieve the goal of advancing the study from merely a philosophical subject to a formal mathematical subject?

A

no

(HAHAHAHA taas kaayo question unya ani ra answer)

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8
Q

Several mathematicans who contributed to the advancement of symbolic logic as a mathematical discipline

A
  • Augustus De Morgan (1806-1871)
  • George Boole (1851-1864)
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9
Q
  • declarative sentence
  • either true or false
  • but not both true and false
A

statement/proposition

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10
Q

To determine whether a sentence is a statement, it may not be necessary to determine whether it is __

A

true

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11
Q

a statement is either true or false but not

A

both true and false

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12
Q

statement that conveys a single idea

A

simple statement

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13
Q

statement that conveys two or more ideas

A

compound statement

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14
Q

denial of a statement

A

negation

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15
Q

if p is any proposition, what is its negation symbol

A

~p (not p)

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16
Q

negation of a true statement

A

false

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17
Q

negation of a false statement

A

true

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18
Q

truth table for negation
p ~p
T
F

A

p ~p
T F
F T

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19
Q

some, all, and no (or none)

A

quantifiers

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20
Q

examples of quantifiers

A
  • some
  • all
  • no (or none)
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21
Q

Negation of:
all p are q

A

some p are not q

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22
Q

Negation of:
some p are q

A

no p are q

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23
Q
  • if p and q denotes to propositions
  • compound proposition “p and q”
A

conjunction of p and q

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24
Q

symbol of conjunction of p and q

A

p ∧ q

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25
Q

a conjunction is true only if

A

both p and q are true

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26
Q

truth table of conjunction of p and q
p q p ∧ q
T T
T F
F T
F F

A

p q p ∧ q
T T T
T F F
F T F
F F F

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27
Q
  • compound proposition
  • formed by joining of two or more propositions by the word “or”
A

disjunction

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28
Q

symbol for disjunction of p and q

A

p ∨ q

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29
Q

equivalent to “and/or”

A

inclusive sense

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30
Q

inclusive sense is equivalent to

A

and/or

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31
Q

equivalent to “either … or” but not both

A

exclusive sense

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32
Q

exclusive sense is equivalent to

A

“either … or” but not both

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33
Q

Logicians have agreed to use the inclusive “or” meaning

A

one or the other or both

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34
Q

truth table of disjunction of p and q
p q p ∨ q
T T
T F
F T
F F

A

p q p ∨ q
T T T
T F T
F T T
F F F

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35
Q

proposition “if p, then q”

A

conditional/implication

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36
Q

symbol for conditional or implication of p and q

A

p => q

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37
Q

truth table for the conditional or implication
p q p => q
T T
T F
F T
F F

A

p q p => q
T T T
T F F
F T T
F F T

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38
Q

if p, then q

A

conditional

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39
Q

conditional

A

if p, then q

40
Q
  • interchaning the antecedent p with consequent q
  • if q, then p
A

converse

41
Q

converse

A

if q, then p

42
Q
  • negating the antecedent p and negating the consequent q
  • if ~p, then ~q
A

inverse

43
Q

inverse

A

if ~p, then ~q

44
Q
  • negating both the antecedent p and the consequent q and interchanging them
  • if ~q, then ~p
A

contrapositive

45
Q

contrapositive

A

if ~q, then ~p

46
Q

truth table for converse
p q q => p
T T
T F
F T
F F

A

p q q => p
T T T
T F T
F T F
F F T

47
Q

truth table for inverse
p q ~p => ~q
T T
T F
F T
F F

A

p q ~p => ~q
T T T
T F T
F T F
F F T

48
Q

truth table for contrapositive
p q ~q => ~p
T T
T F
F T
F F

A

p q ~q => ~p
T T T
T F F
F T T
F F T

49
Q

statement of the form
(p => q) ∧ (q => p)

A

biconditional

50
Q

symbol for biconditional

A

p <=> q

51
Q

how to read the symbol
p <=> q

A

p if and only if q

52
Q

abbreviation of “p if and only if q”

A

p iff q

53
Q

biconditional means

A

two conditionals

54
Q

a biconditional can also be expressed by using __ and __ terminology

A
  • necessary
  • sufficient
55
Q

“p is sufficient for q” can be rephrased as

A

if p then q

56
Q

“p is necessary for q” can be rephrased as

A

if q then p

57
Q

biconditional “p if and only if q” can also be phrased as

A

p is necessary and sufficient for q

58
Q

biconditional combines a __ and its __ into one statement

A
  • conditional
  • converse
59
Q

truth table for biconditional
p q p=>q q=>p q<=>p
T T T T
T F F T
F T T F
F F T T

A

p q p=>q q=>p q<=>p
T T T T T
T F F T F
F T T F F
F F T T T

60
Q

words that connect simple statements and creates a compound statement

A
  • and
  • or
  • if…then
  • if and only if
61
Q

used symbols such as p, q, r, and s to represent statements and symbols ∧, ∨, ~, =>, <=> to represent connectives

A

George Boole

62
Q

when someone makes a sequence of statements and draws some conclusion from them, he or she is presenting an __

A

argument

63
Q

two components of arguments

A
  1. initial statements (hypotheses)
  2. final statement (conclucsion)
64
Q

if the conclusion of the argument is guaranteed under its given set of hypotheses

A

valid

65
Q

consists of a set of statements called premises and another statement called the conclusion

A

argument

66
Q

set of statements in an argument

A

premises

67
Q

another statement in an argument

A

conclusion

68
Q

an argument is __ if the conclusion is true whenever all the premises are assumed to be true

A

valid

69
Q

an argument is __ if it not a valid argument

A

invalid

70
Q

Two Argument Types

A
  1. inductive argument
  2. deductive argument
71
Q

uses a collection of specific examples as its premises and uses them to propose a general conclusion

A

inductive argument

72
Q

inductive argument

A

from specific to general

73
Q

uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion

A

deductive argument

74
Q

deductive argument

A

from general to specific

75
Q

it is customary to place a __ __ between the premises and the conclusion

A

horizontal line

76
Q

inductive or deductive
Premises:
I ate candy last week and I had toothache.
I ate candy yesterday and I had toothache.
I ate candy today, and I had toothache.
——————————————————
Conclusion:
Therefore, eating candy gives me toothache.

A

inductive

77
Q

inductive or deductive
Premises:
all cats are mammals.
a tiger is a cat.
——————————-
Conclusion:
A tiger is a mammal.

A

deductive

78
Q

inductive or deductive
Premises:
all doctors are men.
my mother is a doctor.
———————————
Conclusion:
Therefore, my mother is a man.

A

deductive

79
Q

Truth table procedure to determine validity of argument

A
  1. write the argument in symbolic form.
  2. construct truth table that shows truth value of each premise and truth value of conclusion for all combinations of truth values of simple statements.
  3. if conclusion is true in every row of truth table in which all the premises are true, the argument is valid. if the conclusion is false in any row in which all of the premises are true, the argument is valid.
80
Q

diagram consisting of various overlapping figures contained within a rectangle

A

Venn diagram

81
Q

rectangle of a Venn diagram is called

A

universe

82
Q

depict form of “all A are B” or “if A, then B”

A

two circles, one inside the other
- inner circle = A
- outer circle = B

83
Q

saying that an argument is valid does not mean that the conclusion is __

A

true

84
Q

Standard forms of Four Valid Arguments

A
  1. direct reasoning
  2. contrapositive reasoning
  3. transitive reasoning
  4. disjunctive reasoning
85
Q

direct reasoning symbol

A

p => q
p
———
∴ q

86
Q

contrapositive reasoning symbol

A

p => q
~q
———
∴ ~p

87
Q

transitive reasoning symbol

A

p => q
q => r
———
∴ p => r

88
Q

disjunctive reasoning symbol

A

p V q
~p
——-
∴ q

or

p V q
~q
——-
∴ p

89
Q

Determine valid conclusion:
if julia is a lawyer, then she will be able to help us.
julia is not able to help us.
——————————-
∴

A

Julia is not a lawyer.

90
Q

Determine valid conclusion:
if they had a good time, they will return.
if they return, we will make more money.
———————————-
∴

A

If they had a good time, we will make more money.

91
Q

Determine valid conclusion:
if you can dream it, you can do it.
you can dream it.
————————–
∴

A

you can do it

92
Q

Determine valid conclusion:
i bought a car or i bought a motorcycle.
i did not buy a car.
————————-
∴

A

I bought a motorcycle.

93
Q

Standard Forms of Two Invalid Arguments

A
  1. Fallacy of the converse
  2. Fallacy of the inverse
94
Q

Fallacy of the converse or

A

Fallacy of affirming the conclusion

95
Q

Fallacy of the inverse or

A

Fallacy of denying the hypothesis

96
Q

Fallacy of the converse symbol

A

p => q
q
——-
∴ p

97
Q

Fallacy of the inverse symbol

A

p =>
~p
——-
∴ ~q

Mathematics in the Modern World (13 decks)

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