Module 5. Mathematical Logic

Question 1

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

Answer 1

Logic

Question 2

body of statements or mathematical sentences

Answer 2

Mathematics

Question 3

important in understanding mathematics as a logical system

Answer 3

symbolic language

Question 4

basis of all mathematic reasoning

Answer 4

logic

Question 5

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

Answer 5

Gottfried Wilhelm Leibniz (1646-1716)

Question 6

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

Answer 6

• philosophical
• mathematical

Question 7

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

Answer 7

no

(HAHAHAHA taas kaayo question unya ani ra answer)

Question 8

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

Answer 8

• Augustus De Morgan (1806-1871)
• George Boole (1851-1864)

Question 9

• declarative sentence
• either true or false
• but not both true and false

Answer 9

statement/proposition

Question 10

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

Answer 10

true

Question 11

a statement is either true or false but not

Answer 11

both true and false

Question 12

statement that conveys a single idea

Answer 12

simple statement

Question 13

statement that conveys two or more ideas

Answer 13

compound statement

Question 14

denial of a statement

Answer 14

negation

Question 15

if p is any proposition, what is its negation symbol

Answer 15

~p (not p)

Question 16

negation of a true statement

Answer 16

false

Question 17

negation of a false statement

Answer 17

true

Question 18

truth table for negation
p ~p
T
F

Answer 18

p ~p
T F
F T

Question 19

some, all, and no (or none)

Answer 19

quantifiers

Question 20

examples of quantifiers

Answer 20

• some
• all
• no (or none)

Question 21

Negation of:
all p are q

Answer 21

some p are not q

Question 22

Negation of:
some p are q

Answer 22

no p are q

Question 23

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

Answer 23

conjunction of p and q

Question 24

symbol of conjunction of p and q

Answer 24

p ∧ q

Question 25

a conjunction is true only if

Answer 25

both p and q are true

Question 26

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

Answer 26

p q p ∧ q T T T T F F F T F F F F

Question 27

- compound proposition - formed by joining of two or more propositions by the word "or"

Answer 27

disjunction

Question 28

symbol for disjunction of p and q

Answer 28

p ∨ q

Question 29

equivalent to "and/or"

Answer 29

inclusive sense

Question 30

inclusive sense is equivalent to

Answer 30

and/or

Question 31

equivalent to "either ... or" but not both

Answer 31

exclusive sense

Question 32

exclusive sense is equivalent to

Answer 32

"either ... or" but not both

Question 33

Logicians have agreed to use the inclusive "or" meaning

Answer 33

one or the other or both

Question 34

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

Answer 34

p q p ∨ q T T T T F T F T T F F F

Question 35

proposition "if p, then q"

Answer 35

conditional/implication

Question 36

symbol for conditional or implication of p and q

Answer 36

p => q

Question 37

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

Answer 37

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

Question 38

if p, then q

Answer 38

conditional

Question 39

conditional

Answer 39

if p, then q

Question 40

- interchaning the antecedent p with consequent q - if q, then p

Answer 40

converse

Question 41

converse

Answer 41

if q, then p

Question 42

- negating the antecedent p and negating the consequent q - if ~p, then ~q

Answer 42

inverse

Question 43

inverse

Answer 43

if ~p, then ~q

Question 44

- negating both the antecedent p and the consequent q and interchanging them - if ~q, then ~p

Answer 44

contrapositive

Question 45

contrapositive

Answer 45

if ~q, then ~p

Question 46

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

Answer 46

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

Question 47

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

Answer 47

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

Question 48

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

Answer 48

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

Question 49

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

Answer 49

biconditional

Question 50

symbol for biconditional

Answer 50

p <=> q

Question 51

how to read the symbol p <=> q

Answer 51

p if and only if q

Question 52

abbreviation of "p if and only if q"

Answer 52

p iff q

Question 53

biconditional means

Answer 53

two conditionals

Question 54

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

Answer 54

- necessary - sufficient

Question 55

"p is sufficient for q" can be rephrased as

Answer 55

if p then q

Question 56

"p is necessary for q" can be rephrased as

Answer 56

if q then p

Question 57

biconditional "p if and only if q" can also be phrased as

Answer 57

p is necessary and sufficient for q

Question 58

biconditional combines a __ and its __ into one statement

Answer 58

- conditional - converse

Question 59

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

Answer 59

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

Question 60

words that connect simple statements and creates a compound statement

Answer 60

- and - or - if...then - if and only if

Question 61

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

Answer 61

George Boole

Question 62

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

Answer 62

argument

Question 63

two components of arguments

Answer 63

1. initial statements (hypotheses) 2. final statement (conclucsion)

Question 64

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

Answer 64

valid

Question 65

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

Answer 65

argument

Question 66

set of statements in an argument

Answer 66

premises

Question 67

another statement in an argument

Answer 67

conclusion

Question 68

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

Answer 68

valid

Question 69

an argument is __ if it not a valid argument

Answer 69

invalid

Question 70

Two Argument Types

Answer 70

1. inductive argument 2. deductive argument

Question 71

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

Answer 71

inductive argument

Question 72

inductive argument

Answer 72

from specific to general

Question 73

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

Answer 73

deductive argument

Question 74

deductive argument

Answer 74

from general to specific

Question 75

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

Answer 75

horizontal line

Question 76

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.

Answer 76

inductive

Question 77

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

Answer 77

deductive

Question 78

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

Answer 78

deductive

Question 79

Truth table procedure to determine validity of argument

Answer 79

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.

Question 80

diagram consisting of various overlapping figures contained within a rectangle

Answer 80

Venn diagram

Question 81

rectangle of a Venn diagram is called

Answer 81

universe

Question 82

depict form of "all A are B" or "if A, then B"

Answer 82

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

Question 83

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

Answer 83

true

Question 84

Standard forms of Four Valid Arguments

Answer 84

1. direct reasoning 2. contrapositive reasoning 3. transitive reasoning 4. disjunctive reasoning

Question 85

direct reasoning symbol

Answer 85

p => q p --------- ∴ q

Question 86

contrapositive reasoning symbol

Answer 86

p => q ~q --------- ∴ ~p

Question 87

transitive reasoning symbol

Answer 87

p => q q => r --------- ∴ p => r

Question 88

disjunctive reasoning symbol

Answer 88

p V q ~p ------- ∴ q or p V q ~q ------- ∴ p

Question 89

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

Answer 89

Julia is not a lawyer.

Question 90

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

Answer 90

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

Question 91

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

Answer 91

you can do it

Question 92

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

Answer 92

I bought a motorcycle.

Question 93

Standard Forms of Two Invalid Arguments

Answer 93

1. Fallacy of the converse 2. Fallacy of the inverse

Question 94

Fallacy of the converse or

Answer 94

Fallacy of affirming the conclusion

Question 95

Fallacy of the inverse or

Answer 95

Fallacy of denying the hypothesis

Question 96

Fallacy of the converse symbol

Answer 96

p => q q ------- ∴ p

Question 97

Fallacy of the inverse symbol

Answer 97

p => ~p ------- ∴ ~q