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Young Won Lim3/17/18
Implication (6A)
Young Won Lim3/17/18
Copyright (c) 2015 – 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Logic (6A)Implication 4 Young Won Lim
3/17/18
Material Implication & Logical Implication
Given two propositions A and B,If A ⇒ B is a tautologyIt is said that A logically implies B (A ⇛ B)
Material Implication A ⇒ B (not a tautology)Logical Implication A ⇛ B (a tautology)
A B A⇒B T T TT F FF T TF F T
A B A∧B A∧B ⇒ A
T T T TT F F TF T F TF F F T
A∧B ⇛ A
tautology
in every interpretation
A → BA ⇒ B
Logic (6A)Implication 12 Young Won Lim
3/17/18
Logic and Venn diagram (1)
S N S⇒N T T TT F FF T TF F T
S N
case
case
case
case
Logic (6A)Implication 13 Young Won Lim
3/17/18
Logic and Venn diagram (2)
p q r p→qT T T TT T F TT F T FT F F FF T T TF T F TF F T TF F F T
case case case case case case case case
p q
r
⋯
Logic (6A)Implication 14 Young Won Lim
3/17/18
Material Implication and Venn Diagram
S N S⇒N T T TT F FF T TF F T
When S⇒N is True
S N
S N
S̄ N̄
S N
case case case case
Logic (6A)Implication 15 Young Won Lim
3/17/18
When S N⇒ is a true statement
S N S⇒N T T TT F FF T TF F T
S⇒N is True
if the conditional statement (S⇒N) is a true statement, then the consequent N must be true if S is true the antecedent S can not be true without N being true
(1)
(2)
(1)(2)
S N
cases ++
case case case case
Logic (6A)Implication 16 Young Won Lim
3/17/18
S ⊆ N
if the conditional statement (S⇒N) is a true statement,
then the consequent N must be true if S is true
S⊆N x∈S x∈N
S⇒N
S N
case cases +
Logic (6A)Implication 17 Young Won Lim
3/17/18
~N ⊆ ~S
if the conditional statement (S⇒N) is a true statement,
the antecedent S can not be true without N being true
¬N⇒¬S
¬N⊆¬S x∈¬N x∈¬S
SN
N̄
case cases +
Logic (6A)Implication 18 Young Won Lim
3/17/18
Material Implication vs. Logical Consequence
S⇒N
S N
S
N
S⇛N
Material Implication Logical Consequences
T⇒F exists Always True (Tautology)
S ⊢ N syntactic (proof)
S ⊨ N semantic (model)
entailment
Logic (6A)Implication 19 Young Won Lim
3/17/18
Implication
http://en.wikipedia.org/wiki/
S NS
N
If S, then N.
S implies N.
N whenever S.
S is sufficient for N.
S only if N.
not S if not N.
not S without N.
N is necessary S.
Logic (6A)Implication 20 Young Won Lim
3/17/18
Necessity and Sufficiency
S N
S
N
condition that must be satisfied for S
necessity for S
without N, it can’t be S
S only if N
condition that guarantees N
sufficiency for N
S satisfies at least N N if S
Logic (6A)Implication 21 Young Won Lim
3/17/18
Other Necessary Conditions
http://philosophy.wisc.edu/hausman/341/Skill/nec-suf.htm`
Get A Term Paper submitted
S N
necessary conditions of S
Get ATerm Paper
S only if N
not S if not N
Midterm taken
Final test taken
Rank < 15%
requirement 1
requirement 2
requirement n
Implication (2A) 22 Young Won Lim3/17/18
Other Sufficient Conditions
http://philosophy.wisc.edu/hausman/341/Skill/nec-suf.htm`
score 100 on every work
Get A
S N
sufficient conditions of N
Get ‘A’
N if S
N whenever STerm Paper
score 100 on every work
Case 1Case 1
Case 2
Case 3
Case n
case 1:100, 95, 90, …
case 2:80, 95, 99, …
necessary conditions of Get A
Young Won Lim3/19/18
Resolution (7A)
Young Won Lim3/19/18
Copyright (c) 2015 – 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Logic (7A)Resolution 3 Young Won Lim
3/19/18
Argument
( p∨q)
(¬ p∨ r)
q∨ r ( p∨q) ∧ (¬ p∨ r) → q∨ r
Logic (7A)Resolution 4 Young Won Lim
3/19/18
p q r ( p∨q)∧(¬p∨r ) q∨rT T T T TT T F F TT F T T TT F F F FF T T T TF T F T TF F T F TF F F F F
( p∨q) ∧ (¬ p∨ r) → q∨ r
p q r p∨qT T T TT T F TT F T TT F F TF T T TF T F TF F T FF F F F
p q r ¬p ¬p∨rT T T F TT T F F FT F T F TT F F F FF T T T TF T F T TF F T T TF F F T T
Truth Table
Logic (7A)Resolution 5 Young Won Lim
3/19/18
p q r p∨q ¬p∨r (p∨q)∧(¬p∨r) q∨rT T T T T T TT T F T F F TT F T T T T TT F F T F F FF T T T T T TF T F T T T TF F T F T F TF F F F T F F
( p∨q) ∧ (¬ p∨ r) → q∨ r
Whenever and are true, is truep∨q ¬p∨r q∨r
Interpretation of this truth table
A → BTTTTTTTT
A B
case case case case case case case case
Logic (7A)Resolution 6 Young Won Lim
3/19/18
Venn diagram for
p∨q
p q r p∨qT T T TT T F TT F T TT F F TF T T TF T F TF F T FF F F F
p∨q
case case case case case case case case
Logic (7A)Resolution 7 Young Won Lim
3/19/18
Venn diagram for
¬p∨r
p q r ¬p ¬p∨qT T T F TT T F F FT F T F TT F F F FF T T T TF T F T TF F T T TF F F T T
¬ p∨r
case case case case case case case case
Logic (7A)Resolution 8 Young Won Lim
3/19/18
When is true
When is true and is true
p∨q¬p∨r
p∨q ¬p∨r
(p∨q)∧(¬p∨r)
( p∨q) ∧ (¬ p∨ r)
cases +++
Logic (7A)Resolution 9 Young Won Lim
3/19/18
p∨q¬p∨r q∨r
cases +++ cases +++++⊂(p∨q)∧(¬p∨r) q∨r
When is true, is also true( p∨q) ∧ (¬ p∨ r) q ∨ r
Logic (7A)Resolution 10 Young Won Lim
3/19/18
Argument
http://en.wikipedia.org/wiki/Derivative
p∨q
¬p∨r
q∨r
F∨q
T∨r
q
T∨q
F∨r
r
Case 2: p is trueCase 1: p is false
when p is false,q must be true.
when p is true,r must be true.
Therefore regardless of truth value of p, If both premises hold, then the conclusion is trueq∨r
Logic (7A)Resolution 11 Young Won Lim
3/19/18
Resolution Examples
p∨q
¬p∨r
q∨r
p∨q
¬p∨r
p∨q
¬p∨r
q∨r
p∨q
¬p
q
p∨q
¬p
p∨q
¬p
q
p
¬p∨r
r
p
¬p∨r
p
¬p∨r
r
Logic (7A)Resolution 12 Young Won Lim
3/19/18
Resolution in Prolog
(⋯∨⋯) ∧ (⋯∨⋯∨⋯) ∧ (⋯∨⋯)
Conjunctive Norm Form (CNF) is assumed
the variables A, B, C, D, and E are in conjunctive normal form:
¬ A ∧ ( B C ) ∨ ( A B ) ∨ ∧ ( ¬ B C ¬ D ) ∨ ∨ ∧ ( D ¬ E ) ∨ A B ∨ A ∧ B
The following formulas are not in conjunctive normal form:
¬ ( B C ) ∨ ( A B ) C ∧ ∨ A ( B ( D E ) )∧ ∨ ∧
Logic (7A)Resolution 15 Young Won Lim
3/19/18
Example A
p∨qp∨¬r¬p∨q¬q∨r
qp∨¬r¬q∨r
p∨¬rr
p
James Aspnes, Notes on Discrete Mathematics, CS 202: Fall 2013
(p∨q)∧(p∨¬r)∧(¬p∨q)∧(¬q∨r)
(p∨q)∧(p∨¬r)∧(¬p∨q)∧(¬q∨r )∧(q)
( p∨q)∧(p∨¬r)∧(¬p∨q)∧(¬q∨r)∧(q)∧(r)
(p∨q)∧(p∨¬r)∧(¬p∨q)∧(¬q∨r)∧(q)∧(r)∧(p)
├
├
├
Logic (7A)Resolution 16 Young Won Lim
3/19/18
Example B – (1)
p → q∨rp∨¬qr∨q
¬p∨q∨rp∨¬qr∨q
Discrete Mathematics, Johnsonbough
q∨¬q∨rr∨q
r∨q
q∨¬q∨r = T∨r = T
p → q∨rp∨¬qr∨q
¬p∨q∨rp∨¬qr∨q
p∨r
Logic (7A)Resolution 17 Young Won Lim
3/19/18
Example B – (2)
p → q∨rp∨¬qr∨q
p → q∨r¬p → ¬qq∨r
Discrete Mathematics, Johnsonbough
p → q∨rq → pq∨r
p∨r
Logic (7A)Resolution 18 Young Won Lim
3/19/18
p q r ¬p ¬p∨q∨rT T T F TT T F F TT F T F TT F F F FF T T T TF T F T TF F T T TF F F T T
p q r ¬q p∨¬qT T T F TT T F F TT F T T TT F F T TF T T F FF T F F FF F T T TF F F T T
Truth Table
p q r q∨rT T T TT T F TT F T TT F F FF T T TF T F TF F T TF F F F
p q r ¬p∨q∨r p∨¬q q∨r H 1∧H 2∧H 3 p∨rT T T T T T T TT T F T T T T TT F T T T T T TT F F F T F F TF T T T F T F TF T F T F T F FF F T T T T T TF F F T T F F F
H 1 = ¬p∨q∨rH 2 = p∨¬qH 3 = q∨r
H 1∧H 2∧H 3 → H 3H 1∧H 2∧H 3 → H 2H 1∧H 2∧H 3 → H 1H 1∧H 2∧H 3 → (p∨r )
Young Won Lim3/19/18
References
[1] http://en.wikipedia.org/[2]
Young Won Lim3/16/18
Functions (4A)
Young Won Lim3/16/18
Copyright (c) 2015 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using OpenOffice and Octave.
Functions (4A) 3 Young Won Lim3/16/18
Function
https://en.wikipedia.org/wiki/Function_(mathematics)
A function that associates any of the four colored shapes to its color.
A function f takes an input x, and returns a single output f(x). One metaphor describes the function as a "machine" or "black box" that for each input returns a corresponding output.
Functions (4A) 4 Young Won Lim3/16/18
Function
https://en.wikipedia.org/wiki/Function_(mathematics)
The above diagram represents a function with domain {1, 2, 3}, codomain {A, B, C, D} and set of ordered pairs {(1,D), (2,C), (3,C)}. The image is {C,D}.
However, this second diagram does not represent a function. One reason is that 2 is the first element in more than one ordered pair. In particular, (2, B) and (2, C) are both elements of the set of ordered pairs. Another reason, sufficient by itself, is that 3 is not the first element (input) for any ordered pair. A third reason, likewise, is that 4 is not the first element of any ordered pair.
Functions (4A) 5 Young Won Lim3/16/18
Domain, Codomain, and Range
X : domain Y : domain
range
Functions (4A) 6 Young Won Lim3/16/18
Function condition : One emanating arrow
Each, one starting arrow
Some, no starting arrow
Some, two starting arrows
Function (O) Function (X) Function (X)
Relation (O) Relation (O) Relation (O)
Functions (4A) 7 Young Won Lim3/16/18
Composite Function
https://en.wikipedia.org/wiki/Function_(mathematics)
A composite function g(f(x)) can be visualized as the combination of two "machines". The first takes input x and outputs f(x). The second takes as input the value f(x) and outputs g(f(x)).
Functions (4A) 8 Young Won Lim3/16/18
Injective Function
https://en.wikipedia.org/wiki/Function_(mathematics)
An injective non-surjective function (injection, not a bijection)
An injective surjective function (bijection)
A non-injective surjective function (surjection, not a bijection)
Functions (4A) 9 Young Won Lim3/16/18
Injective Function
https://en.wikipedia.org/wiki/Function_(mathematics)
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness:
it never maps distinct elements of its domain to the same element of its codomain.
every element of the function's codomain is the image of at most one element of its domain.
The term one-to-one function must not be confused with one-to-one correspondence (a.k.a. bijective function), which uniquely maps all elements in both domain and codomain to each other.
Functions (4A) 10 Young Won Lim3/16/18
Surjective Function
https://en.wikipedia.org/wiki/Function_(mathematics)
A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value of f(x) for at least one point x in the domain.
A non-surjective function from domain X to codomain Y. The smaller oval inside Y is the image (also called range) of f. This function is not surjective, because the image does not fill the whole codomain. In other words, Y is colored in a two-step process: First, for every x in X, the point f(x) is colored yellow; Second, all the rest of the points in Y, that are not yellow, are colored blue. The function f is surjective only if there are no blue points.
Functions (4A) 11 Young Won Lim3/16/18
Surjective Functions
Range = CodomainEvery, arriving arrow(s)
Surjective (O) Surjective (X) Surjective (X)
Functions (4A) 12 Young Won Lim3/16/18
Injective Functions
Every, Less than one arriving arrow
Injective (O) Injective (O) Injective (X)
Functions (4A) 13 Young Won Lim3/16/18
Surjective Function
https://en.wikipedia.org/wiki/Function_(mathematics)
A surjective function is a function whose image is equal to its codomain.
a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with f(x) = y.
Functions (4A) 14 Young Won Lim3/16/18
Surjective Functions
https://en.wikipedia.org/wiki/Function_(mathematics)
Surjective composition: the first function need not be surjective.
Another surjective function. (This one happens to be a bijection)
A non-surjective function. (This one happens to be an injection)
Functions (4A) 15 Young Won Lim3/16/18
Types of Functions
https://en.wikipedia.org/wiki/Function_(mathematics)
surjectioninjection
bijection
Young Won Lim3/16/18
References
[1] http://en.wikipedia.org/[2]
Young Won Lim3/5/18
Set Operations (15A)
Young Won Lim3/5/18
Copyright (c) 2017 - 2 018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Set Operations (15A) 3 Young Won Lim3/5/18
Unions
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 4 Young Won Lim3/5/18
Properties of Unions
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 5 Young Won Lim3/5/18
Intersections
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 6 Young Won Lim3/5/18
Properties of Intersections
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 7 Young Won Lim3/5/18
Complements
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 8 Young Won Lim3/5/18
Complements
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 9 Young Won Lim3/5/18
Complements
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 11 Young Won Lim3/5/18
Inclusion and Exclusion
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 13 Young Won Lim3/5/18
De Morgan’s Law
https://en.wikipedia.org/wiki/Set_(mathematics)
Set Operations (15A) 14 Young Won Lim3/5/18
The Same Cardinality
https://en.wikipedia.org/wiki/Set_(mathematics)
Young Won Lim3/5/18
References
[1] http://en.wikipedia.org/[2]
Young Won Lim3/13/18
Matrix (2A)
Young Won Lim3/13/18
Copyright (c) 2015 - 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Functions (4A) 3 Young Won Lim3/13/18
Row Vector, Column Vector, Square Matrix
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 4 Young Won Lim3/13/18
Matrix Notation
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 5 Young Won Lim3/13/18
Matrix Addition
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 6 Young Won Lim3/13/18
Scalar Multiplication
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 7 Young Won Lim3/13/18
Transposition
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 8 Young Won Lim3/13/18
Matrix Multiplication
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 9 Young Won Lim3/13/18
Properties
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 10 Young Won Lim3/13/18
Matrix Types
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 11 Young Won Lim3/13/18
Identity Matrix
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 12 Young Won Lim3/13/18
Inverse Matrix
https://en.wikipedia.org/wiki/Invertible_matrix
Functions (4A) 13 Young Won Lim3/13/18
Inverse Matrix Examples
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 14 Young Won Lim3/13/18
Solving Linear Equations
A set of linear equations
[a bc d ][ xy ] = [ef ]
a x + b y = e
c x + d y = f [ xy ] = [a bc d]
−1
[ef ]
∣a bc d∣ = ad − bc ≠ 0
∣e bf d∣
[ xy ] =1
ad − bc [ d −b−c a ][ef ]
∣a ec f∣
x =∣e bf d∣
∣a bc d∣
y =∣a ec f∣
∣a bc d∣
= de − bf
= af − ce
[ xy ] =1
ad − bc [ de−bf−ce+af ]
Cramer's Rule
If the inverse matrix exists
Determinant (3A) 15 Young Won Lim03/13/2018
Rule of Sarrus (1)
[a11 a12 a13
a21 a22 a23
a31 a32 a33]
a11 a12
a21 a22
a31 a32[a11 a12 a13
a21 a22 a23
a31 a32 a33]
a11 a12
a21 a22
a31 a32
Determinant of order 3 only
Rule of Sarrus
[a11 a12 a13
a21 a22 a23
a31 a32 a33] [
a11 a12 a13
a21 a22 a23
a31 a32 a33]
a11 a12
a21 a22
a31 a32
Copy and concatenate
+ + + – – –
+ a11a22a33 + a12a23a31 + a13a21a32 − a13a22a31 − a11a23 a32 − a12a21a33
Determinant (3A) 16 Young Won Lim03/13/2018
Linear Equations
a11 x1 + a12 x2 + a1n xn = b1+⋯
a21 x1 + a22 x2 + a2n xn = b2+⋯
an1 x1 + an2 x2 + ann xn = bn+⋯
⋮ ⋮ ⋮ ⋮
a11 a12 a1n
a21 a22 a2n
an1 an2 ann
⋮ ⋮⋮
⋯
⋯
⋯
x1
x2
xn
⋮
= b1
= b2
= bn
⋮
(Eq 1)
(Eq 2)
(Eq 3)
Ax = b
Determinant (3A) 17 Young Won Lim03/13/2018
Cramer's Rule (1) – solutions
a11 a12 a1n
a21 a22 a2n
an1 an2 ann
⋮ ⋮⋮
⋯
⋯
⋯
x1
x2
xn
⋮
= b1
= b2
= bn
⋮
A
a11 a12 a1n
a21 a22 a2n
am1 an2 ann
⋮ ⋮⋮
⋯
⋯
⋯
a11 a12a1n
a21 a22a2n
an1 am2ann
⋮ ⋮⋮⋯
⋯
⋯
a11 a12 a1n
a21 a22 a2n
an1 an2 amn
⋮ ⋮⋮
⋯
⋯
⋯
b1
b2
bn
⋮
b1
b2
bn
⋮
b1
b2
bn
⋮
x b
A2A1 An
x1 =det (A1)
det (A)x2 =
det (A2)
det (A)xn =
det (An)
det (A)
Young Won Lim3/13/18
References
[1] http://en.wikipedia.org/[2]