Mathematics for data Science Part 2- Analysis of arguments .pdf

Mathematics for Data Science
UCDS
13-09-2024 Prof. A Choudhury 1

Arguments
Given are two arguments.
1) The local bank was robbed. The robber was
getting away in a FIAT car. Sam is known to have a
record of theft and he owns a FIAT .Therefore, Sam
is the person who robbed the local bank.
2) The local bank was robbed. The person who
robbed the local bank left the only unauthorized
finger print on the safe. Sam’s finger print matches
the unauthorized one left on the safe. Therefore,
Sam is the person who robbed the local bank.

Analysis of arguments
Structure of an argument:
In any argument, there are two parts. The first part consists of evidence or
premises. The second part is the conclusion.
Valid and invalid arguments:
If the premises being true force the conclusion to be true then we say that
the argument is valid.
Otherwise the argument is considered invalid.
Analysis: First Argument:
Premises:
a) The local bank was robbed.
b) The robber was getting away in a FIAT car.
c) Sam is known to have a record of theft and he owns a FIAT .
Conclusion:

Second argument
Analysis: Second Argument:
Premises:
a) The local bank was robbed.
b) The person who robbed the local bank left the only
unauthorized finger print on the safe.
c) Sam’s finger print matches the unauthorized one left
on the safe.
Conclusion:
Which of the two arguments is a valid argument?
How to reason?

1) Symbolize (consistently) all the premises and the conclusion.
This means that we identify the basic statements or input variables
and then express the premises and conclusions in terms of the basic
statements.
2) Create a truth-table for the argument having a column for each
input variable or basic statement appearing in the argument. Further a
column each for the premises and a final column for the conclusion.
3) If there is a row in the truth-table where every entry for the
premises is true but the correspondingentry for the conclusion is
false then such a row is called a bad row.
4)If the truth-table for the argument has a bad row then it is an invalid
argumentotherwise it is a valid argument.

Example 1
Example 1: Consider the following argument:
If I read book A then I will be able to discuss book A in class. I discussed book A in class.
Therefore I read book A.
Basic Statements:
p: I read book A.
q: I discuss book A in class.
Premises:
P1: p → q (If I read book A then I will be able to discuss book A in class.)
P2: q (I discussed book A in class.)
Conclusion: C: p (I read book A.)

Analysis and inference
Analysis:
In the 2nd last row of the table we note that the entries for P1 and P2 are T but the
entry for C is F. It is a bad row.
Inference:
Hence, this is an invalid
argument.
Example 2
I have to see either Movie A or Movie B. I cannot see Movie B.
Therefore I have to see Movie A.
Is this a valid argument?

Patterns
From the analysis we can note certain patterns.
• If any argument can be symbolized in the same pattern as
in Example 1,namely,
P1 : p→ q
P2: q
_________________
C : p
then the argument will have the same truth-table as given in
Table 1 and hence will have a bad row.
Hence, we can conclude that any argument which can be
symbolized in the above pattern is an invalid argument.

Patterns
• Similarly we find that any argument which can be
symbolized in the same pattern as Example 2, namely,
P1 : pꓦ q
P2 : ⁓q
__________
C : p
then the argument will have the same truth-table as given in
Table 2 and so will not have a bad row.
Hence we can conclude that any argument which can be
symbolized in the above pattern is an valid argument.

Valid arguments/valid deduction rule
1)Direct reasoning(a.k.a affirmative mode or modus ponen)
The pattern given below is a valid argument. If an argument in
symbolic form has this pattern then the argument will be valid by
Direct Reasoning.
P1 : p → q
P2 : p
______________
C : q
Example: All cats can climb trees. Kitty is a cat. Therefore Kitty can
climb a tree.
• Note that, Àll cats can climb trees' is the same as the statement
• Ìf it is a cat then it can climb trees'.
• In the above example, let p be the statement Ìf it is a cat' and q
be the statement Ìt can climb trees'.

2) Contrapositive Reasoning(a.k.a Denial mode or modus
tollen)
Any argument that can be symbolized to have the pattern
given below is a valid argument and this valid pattern of
reasoning is called Contrapositive Reasoning.
P1 : p → q
P2 : ⁓ q
_________
C : ⁓ p
Example: All tigers are carnivores. Tom is not a carnivore.
Therefore Tom is not a tiger.

3) Disjunctive Syllogism
Any argument that can be symbolized to have the pattern
given below, is a valid argument and this valid pattern of
reasoning is called Disjunctive Syllogism.
P1 : p ꓦq
P2 : ⁓ q
______________
C : p
Example: Sybil will take her dog for a walk or Sybil will
feed her cat. Sybil could not feed her cat. Therefore, Sybil
will take her dog for a walk.

4) Transitive reasoning
The valid pattern of reasoning given below is called Transitive
Reasoning.
If an argument can be symbolized to have the same pattern as
the one below then we say that it is a valid argument using
Transitive Reasoning.
P1 : p → q
P2 : q → r
_____________
C : p → r
Example: All monkeys are mammals. All mammals look after
their young. Therefore, all monkeys look after their young.

Invalid arguments Or Fallacy
1)Fallacy of the converse:
P1 : p → q
P2 : q
_________
C : p
Example: If I read book A then I will be able to discuss book A in class. I
discussed book A in class. Therefore I read book A.
2)Fallacy of the inverse:
P1 : p → q
P2 : ⁓ p
________________
C : ⁓ q
• Example: All policemen are brave. Joe is not a policeman. Therefore Joe is
not brave.

3) False chain:
P1 : p → q
P2 : r → q
________________
C : p → r
Example:If today is a Government holiday then the University is closed. If
today is a Sunday then the University is closed. Therefore if today is a
Government holiday then today is a Sunday.
4) Disjunctive fallacy:
P1 : p ꓦ q
P2 : p
_____________________
C : ⁓ q
Example: You read detective fiction or you read comics. You read detective
fiction. Hence you do not read comics.
Invalid arguments Or Fallacy

Argument
• We have explored four valid patterns and four invalid patterns .
However, it should be emphasized that many arguments, when
symbolized may not fit any of these eight patterns. In such cases,
the arguments will have to be analyzed for bad rows either using a
truth-table or directly.
Example: I got a scholarship and I got an A grade in Logic and
Reasoning. I am good at logic or I get an A grade in Logic and
Reasoning. Therefore I am good at logic or I do not get a scholarship.
The above argument can be symbolized as follows:
P1 : p ∧ q
P2 : r ∨ q
____________________
C : q ∨∼ p

Beyond propositional logic
Consider “x is prime”. These are not propositions since their truth
value depends on the input value of x.
To deal with these situations we use ‘Predicate Logic”.
Predicatelogic extends propositional logic.
Predicate: A predicate is a declarative sentences whose T/F value
depends on one or more variables.
Example: P(x) = “ x is even”
Q(x, y)= ‘x is greater than y”
Predicates are not statements, because they containfree variables.
Quantifier: A quantifier modifies a predicate by describing whether
some or all elements of the domain(universeof discourse) of the
predicate satisfy the predicate.
∀ “for all” (universal quantifier)
∃ “there exists”,“for some” existential quantifier)

Predicate Logic
Universal quantifier: It states that the statements within its scope are true for
every value of the specific variable. It is denoted by ∀.
(∀ 𝑥 ∈ 𝐴), F(x) means ‘For every value of x in the set A , F(x) is true.’
Concretely, given the predicate F(x) : x.0=0 for real numbers,
(∀ x ∈ 𝑹), F(x) means “ For all real numbers x.0=0”.
Universe of discourse is the domain.
Existential quantifier: It states that the statements within its scope are true
for some value of the specific variable. It is denoted by ∃.
Given the predicate R(x) : x-2=0 for an integer.
(∃x)R(x) means “There is some integer x for which x-2=0”.
Universe of discourse is some integer.(domain)

Nested(Compound) quantifiers
Rule 1: If we use the same quantifier , then the ordering doesn’t
matter.
• ‘For all x ∈ R and for all y ∈ R, x + y = 4.’, is the same as
‘For all y ∈ R and for all x ∈ R, x + y = 4.’, which is the same as
‘For all x, y ∈ R, x + y = 4.’ (Note: You should be able to tell that this is
a false statement.)
• ‘There exists x ∈ R and there exist y ∈ R such that x + y = 4.’, is the
same as
‘There exists y ∈ R and there exists x ∈ R such that x + y = 4.’, which is
the same as
‘There exist x, y ∈ R such that x + y = 4.’ (Note: You should be able to
tell that is a true statement.)

Rule 2:If we are using mixed quantifiers , the order matters.
Let 𝐺(𝑥 , 𝑦) represent the predicate “𝑥 > 𝑦”.
a) ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ 𝐵 , 𝐺(𝑥, 𝑦)(quantified predicate)
“For all numbers y ∈ 𝐴 , there exists some number x ∈ 𝐵 such
that 𝑥 > 𝑦 is true”.
This is true statement.
b) ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 , 𝐺(𝑥, 𝑦)
“There exists some number 𝑥 ∈ 𝐴 such that for all numbers
y ∈ 𝐵, x>y”.
This is false statement. Why?
Nested(Compound) quantifiers

Negation rules
Negation Rules: When we negate a quantified statement, we negate all the
quantifiers first, from left to right (keeping the same order), then we negative
the statement.
1. ∼ ∀𝑥 𝑃 𝑥 ≡ ∃𝑥 [~𝑃 𝑥 ]
2. ~ ∃𝑥 𝑃 𝑥 ≡ ∀𝑥 [~𝑃 𝑥 ]
3. ~ ∀𝑥 ∃𝑦 𝑃 𝑥, 𝑦 ≡ ∃𝑦 ∀𝑥 [~𝑃 𝑥, 𝑦 ]
4. ~[ ∃𝑥 ∀𝑦 𝑃 𝑥, 𝑦 ] ≡ (∀𝑥)(∃𝑦)[~𝑃 𝑥, 𝑦 ]

Examples
1. 𝐿𝑒𝑡 𝑃 𝑥 : 𝑥2
= 2.
𝐼𝑓 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑥 𝑖𝑠 𝒁 𝑜𝑟 𝑵 𝑡ℎ𝑒𝑛 𝑃 𝑥 𝑖𝑠 𝑛𝑜𝑡 𝑡𝑟𝑢𝑒 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥.
Therefore, the statement ∃𝑥 ∈ 𝒁 𝑃 𝑥 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒 , ∀𝑥 ∈ 𝑵 𝑃 𝑥 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒.
𝐵𝑢𝑡 𝑖𝑓 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑥 𝑖𝑠 𝑹, 𝑡ℎ𝑒𝑛 ∃𝑥 ∈ 𝑅, 𝑃 𝑥 is true.
2. 𝐿𝑒𝑡 𝑃 𝑥, 𝑦 𝑏𝑒 𝑡ℎ𝑒 𝑝𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛: 𝑥 < 𝑦.
𝑈𝑠𝑒 𝑃 𝑥, 𝑦 𝑎𝑛𝑑 𝑞𝑢𝑎𝑛𝑡𝑖𝑓𝑖𝑒𝑟𝑠 𝑡𝑜 𝑒𝑥𝑝𝑟𝑒𝑠𝑠 𝑡ℎ𝑒 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡: “𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡
𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟.
” 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑞𝑢𝑎𝑛𝑡𝑖𝑓𝑖𝑒𝑟 𝑦𝑜𝑢 𝑢𝑠𝑒?

Negation rules for predicate logic
Domain of
people
P(x) “x is a liar”
1 (∀𝑥)𝑃(𝑥) All people are liars.
~[(∀𝑥)𝑃(𝑥)] Negation
(∃𝑥)[~𝑃(𝑥)] There exists a person who is not a liar
2 (∃𝑥)𝑃(𝑥) There exists a person who is a liar
~[(∃𝑥)𝑃(𝑥)] Negation
(∀𝑥)[~𝑃(𝑥)] There are no liars

Mathematical logic
Mathematical statements:
• Definition
• Postulates or axioms
• Theorem- corollary, proposition, lemma
Definition : It is a statement that stipulates the meaning of a new term,
symbol or an object.
Example:
Definition of parallel lines: Two lines in a plane are parallel if they do not
have a common point.
• Mathematical definitions are extremely precise.
• They are logically “ if and only if” statements.
• In predicate logic, if D(x)= x is (defined term)
P(x)= [defining property of the term]
then a definition is of the form (∀𝑥)(𝐷(𝑥) 𝑃(𝑥)) .

Mathematical statements
Axioms or postulates: Propositionsthat are accepted to be true are called axioms or
postulates.
Example: “There is a straight line segment between a pair of points”.
• Axioms are basic , fundamental true statements.
• Theorems are based on the assumption of some certain sets of underlying axioms.
Euclid established the truth of many additional propositions by providing ‘ proofs’.
Theorem:A theorem is a statement that follows logically from statements we have
already established or taken as given.
Proof:A proof is a sequence of logical deductions from axioms and previously proved
statements that concludes with the proposition in question.
A lemma is a preliminary proposition useful for proving later propositions.
A corollary is a proposition that follows in just a few logical steps from a theorem.
• A statement that we intend to prove is called a claim.
• A statement that we can’t yet prove but that we suspect is true is called a
conjecture.

Axiomatic approach
• Euclid’s axioms and proof approach
• Foundation of mathematics
• ZFC axioms together with a few logicaldeduction rules,
appear to be sufficient to derive essentially all of
mathematics.
Example:
The axioms of Euclidean Plane geometry.
Counterexample:A particular value that shows a statement to
be false is called a counterexample to the statement.
Example: “ Every prime number is an odd number”.
Counterexample: 9 is not a prime.

The Axioms of Euclidean Plane geometry.
1. A straight line may be drawn between any two points.
2. Any terminated straight line may be extended
indefinitely.
3. A circle may be drawn with any given point as center
and any given radius.
4. All right angles are equal.
5. If two straight lines in a plane are met by another line,
and if the sum of the internal angles on one side is less
than two right angles, then the straight lines will meet if
extended sufficiently on the side on which the sum of the
angles is less than two right angles.

Fractals

Well known theorems and conjectures
1. Euler’s Conjecture
2. Four colour theorem
3. Goldbach’s conjecture
4. Fermat’s last theorem
5. Claim: The equality 313(x3+y3)=z 3 , x, y, z∈ Z +
has no solution.

Proving an “implication”
“ If p then q.”
Example1:
If 0≤x≤ 2, then –x3 +4x+1 >0.
Example 2:
If r is irrational then √r is irrational.
Method #1:Direct proof
Method #2:Prove the contrapositive

Proving ”if and only if”
“If p then q and if q then p” .
Example1:
Two triangles have the same side lengths if and only
if two side lengths and the angle between those
sides are the same.
Example 2:
For every integer n, 6|n if and only if 3|n and 2|n.
Method #1:Prove Each Statement Implies the Other
Method#2: Construct a Chain of iff arguments

Other methods of proofs
• Proofs by cases
• Proof by contradiction
Example:
1.Prove that √3 is an irrational number.
2. If x is a real number such that
𝑥2−1
𝑥+2
>0, then
either x>1 or -2<x<-1.

Annexure A- logical equivalences
1. ~𝑝 ∨ 𝑞 ≡ 𝑝 → 𝑞 (𝑖𝑚𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛)
2. ~ ~𝑝 ≡ 𝑝( 𝑑𝑜𝑢𝑏𝑙𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑜𝑛)
3. ~ 𝑝 → 𝑞 ≡ 𝑝 ∧ ∼ 𝑞 𝑛𝑒𝑔𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖𝑚𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛
4. ∼ 𝑝 ∧ 𝑞 ≡∼ 𝑝 ∨∼ 𝑞 ; ∼ 𝑝 ∨ 𝑞 ≡∼ 𝑝 ∧∼ 𝑞 𝐷𝑒𝑀𝑜𝑟𝑔𝑎𝑛′
𝑠 𝑙𝑎𝑤𝑠
5. 𝑝 ∨ 𝑝 ≡ 𝑝; 𝑝 ∧ 𝑝 ≡ 𝑝 𝐿𝑎𝑤 𝑜𝑓 𝑖𝑑𝑒𝑚𝑝𝑜𝑡𝑒𝑛𝑐𝑒
6. 𝑝 ∨ 𝑝 ∧ 𝑞 ≡ 𝑝; 𝑝 ∧ 𝑝 ∨ 𝑞 ≡ 𝑝 𝐿𝑎𝑤 𝑜𝑓 𝑎𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛
7. 𝑝 ∨ 𝑞 ≡ 𝑞 ∨ 𝑝; 𝑝 ∧ 𝑞 ≡ 𝑞 ∧ 𝑝 𝐶𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒 𝑙𝑎𝑤
8. 𝑝 ∨ 𝑞 ∨ 𝑟 ≡ 𝑝 ∨ 𝑞 ∨ 𝑟; 𝑝 ∧ 𝑞 ∧ 𝑟 ≡ 𝑝 ∧ 𝑞 ∧ 𝑟 𝐴𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑣𝑒 𝑙𝑎𝑤
9. 𝑝 ∨ 𝑞 ∧ 𝑟 ≡ 𝑝 ∨ 𝑞 ∧ 𝑝 ∨ 𝑟 ;
𝑝 ∧ 𝑞 ∨ 𝑟 ≡ 𝑝 ∧ 𝑞 ∧ 𝑝 ∧ 𝑟 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑣𝑒 𝑙𝑎𝑤

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