Logic and Proof

Question 1

What is a proposition

Answer 1

A declarative statement that is either true or false

Question 2

what is this symbol ¬

Answer 2

p turns to ¬p
p = apple is red
¬p = apple is not red

Question 3

what is a truth table

Answer 3

all combinations on the left side on the right side the connectives

Question 4

what is this symbol ∧

Answer 4

conjunction
p∧q = T if both p and q are True.
Both propositions have to be true.

Question 5

what is this symbol ∨

Answer 5

disjunction
p ∨ q = T if p or q is true
one or more statements have to be true

Question 6

what is this symbol ∀

Answer 6

universal quantifier
for all x within the given domain satisfy a predicate to give true

Question 7

what is this symbol →

Answer 7

implication
if p then q or p implies q
hypothesis is true conclusion must be true for the implication to be true.

Question 8

what is one directional implication

Answer 8

in that case of p → q = T
where p = F and q = T, this is telling us the hypothesis is false so the implication will always be true
all numbers divisble by 4 are even
all even numbers arent divisible by 4

Question 9

what is meant by converse

Answer 9

if p→q, the converse is q→p
switch the order.

Question 10

what is meant by inverse

Answer 10

if p→q the inverse would be ¬p→¬q
negate both propositions

Question 11

what is meant by contrapositive

Answer 11

if p→q the contrapositive would be ¬q→¬p
switch and negate

Question 12

what is meat by biconditional

Answer 12

For a biconditional to be true both propositions must share the same truth value
p ↔ q = T if P = T and Q = T or P = F and Q = F

Question 13

what is this symbol ↔

Answer 13

biconditional

Question 14

how do you demonstrate equivalencies between different compound propisitions

Answer 14

truth tables

Question 15

where is the ↔ in the order of precedence

Answer 15

5

Question 16

where is the ¬ in the order of precedence

Answer 16

1

Question 17

where is the → in the order of precedence

Answer 17

4

Question 18

where is the ∨ in the order of precedence

Answer 18

3

Question 19

where is the ∧ in the order of precedence

Answer 19

2

Question 20

what is the definition of a negation

Answer 20

let p be a proposition, negation of p denoted by ¬p is the statement “it is not the case that p”

Question 21

What are the key feature of the negation operator

Answer 21

• first in operator precedence
• not systematically defined symbol
• constructs new propositon based on an existing one

Question 22

Definition of AND

Answer 22

let p and q be propositions, conjunction of p and q denoted by p∧q, is the proposition “p and q”,
* only true when both propositions are true otherwise false

Question 23

What terminologies are utilised in natural language for the AND connective

Answer 23

• but, when the propositions are related
• it is sunny but it is raining , where the two predicates are seemingly unlikely to occur in the same situation or pose a seemingly paradoxical situation.

Question 24

What are key feature of the AND operator

Answer 24

• 2nd in the order of precedence
• only true when both conditions are true otherwise false
• a connective, joining two propositions to form one

Question 25

What is the definition for OR

Answer 25

let p and q be propositions, disjunction of p and q is denoted by p ∨ q , read as "p or q" * false when both propositions truth value is negative

Question 26

key features of disjunction operator ∨

Answer 26

* 3rd in the order of precedence * true if either proposition is true * form new proposition based existing two that dont have to be distinct but would be pointless otherwise.

Question 27

What is the inclusive OR

Answer 27

This is referred to as the inclusive disjunction, such that it would result in true if both or one of the propositions are met

Question 28

What is an example of using the Inclusive OR

Answer 28

p : Students have taken calculus q : Students have taken introductory cs students who have taken calculus or introductory cs will take this class * here if the student has taken one or both they will be eligibile to take the cs class

Question 29

What is the defintion of XOR

Answer 29

Let p and q be propositions, exclusive disjunction between p and q, p ⊕ q, is read as "p or q not both" or "p exc or q", only true when exactly one proposition is true otherwise false.

Question 30

What is the definition of Implication

Answer 30

Let p and q be propositions, the conditional statement p→q is false when p is true and q is false otherwise is true

Question 31

What is a good analogy to consider when thinking of the implication operator

Answer 31

You should think of this operator in the form of a pledge, consier the example if i win an election then i will lower tax. If i win election and i dont lower tax the pledge is broken. Otherwise it is possible i dont win an election and convince others to lower tax. If i dont win the election no one will expect me to lower tax hence this is still true.

Question 32

in the statement p→q what is p and q

Answer 32

* P is the hypothesis, antedecent or premesis * Q is the conclusion, consequence implication

Question 33

What are forms that the statement p→q appears in natural language

Answer 33

* "p is sufficient for q" * "p only if q" * "a sufficient condition for q is p" * "q is necessary for p"

Question 34

Key features of implication operator

Answer 34

4th in order of precedence false if p is true and q is false

Question 35

What is the distinction between mathematics and natural language often when utilising the implication operator

Answer 35

In mathematics it is not required to have a cause and effect relationship between the propositions being implied as we are only concerned with the truth values of the propositions mathematically. natural language would be superficially attached to the propositional objects.

Question 36

what is a contrapositive statement of p→q equal to

Answer 36

it is equal to the original statement p→q

Question 37

What is the converse of p→q equal to

Answer 37

the inverse

Question 38

What is the inverse of p→q equal to

Answer 38

the converse

Question 39

What is p→q equal to in terms of contrapositive, converse and inverse

Answer 39

contrapositive

Question 40

What is the definition of bi-conditional

Answer 40

let p and q be propositions, the biconditional statement, p ↔ q, p if and only if q, only true if both proposition truth values are equal

Question 41

What is this symbol ↔

Answer 41

Equivalence or bi conditional operator

Question 42

What are uses for the biconditional operator in natural language

Answer 42

p if and only if q p is a necessary and sufficient condition for q

Question 43

Feature of the biconditional operator

Answer 43

5th in the order of precedence true if both proposition truth values are the same equivalent to p→q ∧ q→p

Question 44

What is meant by the implicit use of the biconditional operator

Answer 44

In the english language this is often implied such as if you eat your dinner then you can have dessert, we mean if you eat your dinner then and only then you may have dessert. having dinner is a necessary and sufficient condition of having dessert. no dinner no dessert. Hence the use of then and only then.

Question 45

meaning of a bit

Answer 45

binary digit, that can have two possible states * term coined by john tukey, similarly termed software

Question 46

bit string

Answer 46

a sequence of zero or more bits length of the string is the number of bits contained within it.

Question 47

What operators can be perfomed on bit strings

Answer 47

OR, AND XOR