Math Logic study notes


update 2013-02-18  index:




<a name="a056_001">
2012-11-29-10-48 start 
This page 
http://freeman2.com/tute0056.htm
is Liu,Hsinhan self study notes. 
Study topic is Mathematical logic.
No one proof read, this 
page may contain error!

<a name="a056_002">
2012-06-21-11-10 Liu,Hsinhan access
http://www.uow.edu.au/~bmaloney/wuct121/LogicTeacher.pdf
06/21/2012 11:10 AM 831,293 LogicTeacher.pdf

2012-06-21-11-34 Liu,Hsinhan access
http://schmidt.ucg.ie/ma184/screen-4x4.pdf
06/21/2012 10:36 AM 357,318 schmidt.ucg.ie_ma184_screen-4x4.pdf

<a name="a056_003">
2012-07-18-11-01 Liu,Hsinhan access
http://www.abstractmath.org/MM/MMConditional.htm
07/18/2012 10:03 AM 212,330 abstractmath.org_MM_MMConditional.htm

2012-07-18-11-09 Liu,Hsinhan access
http://mysite.verizon.net/thelogos/Logic_Notes01.pdf
07/18/2012 10:11 AM  57,185 mysite.verizon.net-thelogos-Logic_Notes01.pdf

<a name="a056_004">
2012-07-18-11-18 Liu,Hsinhan access
http://www.ksmodern1.com/Logic%20part%203.pdf
07/18/2012 10:21 AM 283,674 ksmodern1.com_Logic%20part%203.pdf

2012-11-23-10-15 Liu,Hsinhan access
http://www.math.umn.edu/~jodeit/course/ACaRA01.pdf
11/23/2012 10:15 AM  41,216 ACaRA01.pdf

<a name="a056_005">
2012-11-23-10-32 Liu,Hsinhan access
http://faculty.swosu.edu/michael.dougherty/book/chapter01.pdf
11/23/2012 10:32 AM 588,180 chapter01.pdf

<a name="a056_006">
Above files are LiuHH's textbook.
The following is in study time sequence.
It is not well organized as textbook did. 

A logic statement is an equation composed 
of logic variables and logic operators.
logic variable itself is simplest logic 
statement.
Example of logic variables: M,N,O .
Example of logic statement: (M∨N)→O .
Assume P is first logic statement and 
assume Q is second logic statement. 
Each statement can have Boolean value 
true or false. Ambiguity (maybe true, 
maybe false) is not allowed.
"P logic operator Q" is a compound statement.

<a name="a056_007">
Logic operation on P,Q has the following.
~P, P∧Q, P∨Q, P⊻Q, P→Q, P↔Q

1. Logical negation '~', not P or ~P
If P=true, ~P=false.
If P=false, ~P=true.
not P is an unitary operation. That is 
operator '~' manipulate just one logic 
statement, not two. ~P is correct, 
~Q is correct, P~Q is WRONG.

Above is unitary operation.
<a name="a056_008">
Below is binary operation.
2012-11-29-11-06 here

2. Logic conjunction, (AND operation)
Assume P,Q are two logic statement.
logic conjunction is P∧Q.
Here '∧' is a math symbol for logic 
conjunction operation. 
Let P be 3>0 which is true
Let Q be 1>0 which is true
then P∧Q is "3>0 AND 1>0"
That is "true AND true" be true.

<a name="a056_009">
Let P be 3>0 which is true
Let R be 1>2 which is false
then P∧R is "3>0 AND 1>2"
That is "true AND false" be false.

Let S be 3>5 which is false
Let R be 1>2 which is false
then S∧R is "3>5 AND 1>2"
That is "false AND false" be false.

If (M AND N) be true, in our mind 
we expect both M be true and N be 
true.
2012-11-29-11-28 here

<a name="a056_010">
Define a general logic truth table as next
Table 01 //a111291130
PQ  logic operation
TT  true or false
TF  true or false
FT  true or false
FF  true or false
<a name="a056_011"> P is first logic statement and Q is second logic statement. P can be true or false, Q can be true or false. Left red column record possible Boolean value for P. Two true and two false. <a name="a056_012"> Second column from left end record possible Boolean value for Q. P two true, one meet Q true, other meet Q false. P two false, one meet Q true, other meet Q false. This accomplish all possible P,Q combination. <a name="a056_013"> Next is AND operation truth table Table 02 //a111291146
PQ  P∧Q comment
TT  true true AND true is true
TF  false true AND false is false
FT  false false AND true is false
FF  false false AND false is false
<a name="a056_014"> Above table comments comply with our common sense. AND mean we want both be true. 2012-11-29-12-00 here <a name="a056_015"> 3. Logic disjunction, (OR operation) Assume P,Q are two logic statement. logic disjunction is P∨Q. Here '∨' is a math symbol for logic disjunction operation. <a name="a056_016"> Next is OR operation truth table Table 03 //a111291203
PQ  P∨Q comment
TT  true true OR true is true
TF  true true OR false is true
FT  true false OR true is true
FF  false false OR false is false
<a name="a056_017"> Above table comments comply with our common sense. OR mean we want either one be true, or both be true. 2012-11-29-12-05 stop <a name="a056_018"> 2012-11-29-16-47 start Example of OR statement is next. Grade A student OR grade B student is a good student. Now, a grade A student stand here, also a grade B student stand here, both be true, above statement is still true. <a name="a056_019"> 4. Logic Exclusive Disjunction (XOR operation) Next is a different OR statement. I study hard, target at get grade A OR get grade B. This is also an OR statement, but it is impossible that grade A be true, grade B be true at same time. This is Exclusive Disjunction (XOR) P ⊻ Q <a name="a056_020"> XOR operation truth table is next Table 04 //a111291659
PQ  P⊻Q comment
TT  false true XOR true is false
TF  true true XOR false is true
FT  true false XOR true is true
FF  false false XOR false is false
<a name="a056_021"> 5. Logic Conditional if P then Q, P→Q P is a simple logic statement true/false. Q is a simple logic statement true/false. if P then Q is a compound statement. A compound statement is itself a logic statement. (ACaRA01.pdf page 4/15) That is Compound statement has true/false value. if P then Q express as P→Q P is assumption, Q is conclusion. <a name="a056_022"> When compound "if P then Q" is true? Assume P and Q both be true, then compound "if P then Q" is true. <a name="a056_023"> Assume P is true and Q is false, then compound "if P then Q" is false. Since if-then structure guarantee that when assumption P is true, the conclusion Q must be true. Now Q is false, promise failed. so "if true then false" is false. <a name="a056_024"> Assume P is false and Q is false, then compound "if P then Q" is true. We can argue that false assumption P imply false conclusion. compound "if P then Q" did not cheat us. Mark it be true. <a name="a056_025"> "if P then Q" last possibility, assume P is false and Q is true, then compound "if P then Q" is true? or false? All book say compound statement "if false then true" is true. explain <a name="a056_026"> Logic Conditional truth table is next. Table 05 //a111291750
PQ  P→Q comment
TT  true if true, result true, OK, P→Q true
TF  false if true, result false, cheating, false.
FT  true "if false then true" is true? why?
FF  true if false result false OK P→Q true
<a name="a056_027"> 2012-07-18-09-58 Liu,Hsinhan study LogicTeacher.pdf page 24/231 to page 27/231 study notes record [[ <a name="a056_028"> he say //page 25/231 P Q P⇒Q T T T T F F F T T <==***** other author is correct F F T <a name="a056_029"> I think P Q P⇒Q T T T T F F F T F <==***** LiuHH is wrong F F T ]] <a name="a056_030"> Liu,Hsinhan think "if false then true" is either trouble or undefined. compound statement "if false then true" should not be true. That is 2012-07-18 to 2012-11-26 LiuHH error thought and puzzle. <a name="a056_031"> 2012-11-27 LiuHH find a solid argument which support "if false then true" be true. This understanding is main motivation of writing tute0056.htm LiuHH will explain later after done "if and only if" compound statement. 2012-11-29-17-36 here <a name="a056_032"> 6. Logic Biconditional P if and only if Q, short symbol is P↔Q P is a simple logic statement true/false. Q is a simple logic statement true/false. P if and only if Q is a compound statement and it is two way. (P→Q is one way) <a name="a056_033"> P if and only if Q suggest that P and Q are equivalent. Therefore, if P=true, Q=true compound P↔Q is true. if P=true, Q=false compound P↔Q is false. if P=false, Q=true compound P↔Q is false. if P=false, Q=false compound P↔Q is true. <a name="a056_034"> P=true, Q=true. P↔Q is true and P=false Q=false P↔Q is true say that P and Q are equivalent. <a name="a056_035"> P=true, Q=false, P↔Q is false and P=false, Q=true, P↔Q is false say that P and Q are not equivalent (one true, another false) then the Biconditional P↔Q is a cheat statement and P↔Q is false. <a name="a056_036"> Logic Biconditional truth table is next. Table 06 //a111291800
PQ  P↔Q comment
TT  true P true, Q true, OK, P,Q equivalent
TF  false P true, Q false P,Q are not equal
FT  false P false, Q true P,Q are not equal
FF  true P false, Q false, OK, P,Q equivalent
<a name="a056_037"> All possible logic binary relation are defined. binary relation is simplest compound statement. It is possible to combine several simple one to form complicated compound statement. For example (P→(Q∨R))∧P∧(~R) see chapter01.pdf page 39/72 2012-11-29-18-08 here <a name="a056_038"> Above five truth tables all have same P/Q true/false value, but different operation get different compound T/F value. "five truth tables" is it complete? Answer is not complete. Five truth tables use P/Q two (n=2) independent variables. There are 2^n=2^2=4 P/Q T/F combinations. Each T/F combination has compound value be true or false. Total possible combination is 2^(2^n)=2^(2^2)=2^4=16 possibilities. More than half of these 16 combinations are dull, P op Q always = true ; or null, P op Q always = false ; or useless, P op Q always = P . Next is <a name="a056_039"> Logic Binary Total Table. 理則二元全表 Table 07 //a111291815
PQ  dulP∨Qfulgul P→QhulP↔QP∧Q vulP⊻Qtulsul rulqulpulnul
TT  TTTT TTTT FFFF FFFF
TF  TTTT FFFF TTTT FFFF
FT  TTFF TTFF TTFF TTFF
FF  TFTF TFTF TFTF TFTF
pq  duliorfulgul ifthuliffand vulxortulsul rulqulpulnul

2012-12-03-04-48 add column name at last row. <"a056_039b"> Please move up a little bit. 2012-12-02-12-30 start Left two columns are elements P/Q table. Column dul to column P∧Q are True Table. Column vul to column nul are False Table. ~(True Table) = False Table. They are dependent, no room for inventing anything. "All things considered" 2012-12-02-12-37 stop

<a name="a056_040"> dul from dull, nul from null. Other *ul follow alphabet sequence. Binary means only P,Q two elements, no third (not P,Q,R three elements) 2012-11-29-18-36 here nul to vul all have P=true,Q=true;P*Q=false they may not be useful (but P⊻Q is defined). nul is always false, nul is not useful. <a name="a056_041"> dul has P=T/F,Q=T/F;P*Q=true always true? dul is not useful. gul always follow P, and Q is ignored, gul is not useful. hul always follow Q, and P is ignored, hul is not useful. <a name="a056_042"> There is one possible logic operation ful and not defined. Read Total Table carefully, we find ful is twin of P→Q P ful Q is in fact Q→P. Total Table tell us that possible logic relations are all used. Invent? try nul to vul. 2021-11-29-18-55 stop //alert, there is no room for Invent. //because ~(nul to vul) is exactly //Logic Binary True Table False Table //and True Table are dependent. //a112021226 <a name="a056_043"> 2021-11-29-21-25 start Now discuss "if false then true" is true? why? From Logic Binary Total Table extract P→Q and P↔Q as following <a name="a056_044"> P→Q, P↔Q table Table 08 //a111292129
PQ  P→Q P↔Q
TT  T T
TF  F F
FT  T F
FF  T T
<a name="a056_045"> On 2012-07-18-09-58 Liu,Hsinhan study notes say above bold red T should be F. //alert! above line is wrong ! From above P→Q, P↔Q table, we know 2012-07-18-09-58 Liu,Hsinhan argument is wrong, because if change T to F, then P→Q and P↔Q two different operation get identical logical result, which is not allowed. 2012-11-29-21-41 here <a name="a056_046"> Following is LiuHH 2012-11-27 argument, which is correct. First of all, P→Q has four possible way to define its result logic value. Although four choices, but only one is correct. Correct choice get consistency everywhere. Wrong choice, conflict will show up. <a name="a056_047"> P→Q,1 to P→Q,4 table Table 09 //a111292148
PQ  P→Q,1 P→Q,2  P→Q,3 P→Q,4
TT  T T  T T
TF  F F  F F
FT  T T  F F
FF  T F  T F
P→Q,1 is correct, other three are wrong. <a name="a056_048"> P→Q,1 to P→Q,4 first row P=T and Q=T, second row P=T and Q=F, do not cause confusion. They all have the compound value first row P→Q true, and second row P→Q false. Third row is a puzzle. <a name="a056_049"> 2012-11-29 use P→Q,1 to P→Q,4 2012-11-27 use four colors, they match as following P→Q,1 red This is correct choice. P→Q,2 grey wrong choice and puzzle. P→Q,3 Purple wrong choice. P→Q,4 blue wrong choice. <a name="a056_050"> P→Q,2 coincide with Q, ignored P, wrong. P→Q,3 conflict with P↔Q, wrong. P→Q,4 conflict with P∧Q, wrong. Only P→Q,1 is correct. <a name="a056_051"> Following has another reason to choose P→Q,1 out of four possibilities. Table 10 (three tables) //a111292217
QP  Q→P1
TT  T
TF  F
FT  T
FF  T
PQ  P→Q
TT  T
TF  F
FT  T
FF  T
PQ  Q→P2
TT  T
TF  T
FT  F
FF  T
<a name="a056_052"> Above Q→P1 and P→Q have same compound result T,F,T,T. but different P,Q pair value. For example left table 2nd row has P=F,Q=T middle table 2nd row has P=T,Q=F Logic variables P=T/F, Q=T/F are different, then we can not compare compound T/F result. <a name="a056_053"> Adjust left table to right table, such that middle table and right table have same P,Q pair value but different compound result. After this adjustment, P→Q and Q→P2 have same P,Q pair value. Which allow us to compare two compound result correctly. <a name="a056_054"> Next is 2012-11-27 Liu,Hsinhan argument. This explanation is correct. 2012-11-29-22-41 here <a name="a056_055"> 2012-11-27-21-21 start //two days earlier Table 11 (two tables) //a111272122
pq  pq
TT   T1 
TF   F3 
FT   T5 
FF   T7 
pq  pq
TT   T2 
TF   T4 
FT   F6 
FF   T8 
<a name="a056_056"> Key: Apply m ∧ n to p → q and p ← q get p ↔ q, correct answer m == (p → q) ; n == (p ← q) Table 12 (two tables) //a111272133
mn  mn
TT   T 
TF   F 
FT   F 
FF   F 
pq  p→qp←qM∧N pq
TTgEt T1T2T1∧T2 =T 
TFgEt F3T4F3∧T4 =F 
FTgEt T5F6T5∧F6 =F 
FFgEt T7T8T7∧T8 =T 
"gEt" apply Table 11 result. "M∧N" is "(p→q)∧(p←q)" get p↔q AND operation "M∧N" get iff result p↔q because Table 11 T4 and T5 are defined correctly. If define wrong, no such luck. <a name="a056_057"> Above false∧false=F and false↔false=T determine the correct choice of false→true=T P→Q four possibilities, the correct one applied to above tables. Top two tables are P→Q and Q→P (p ← q) table stand on equal P,Q pair value. 2012-11-29-22-46 here <a name="a056_058"> Apply m ∧ n to p → q and to p ← q get p ↔ q table. Here m is (p → q) and n is (p ← q) Above red T are "vacuously true" or defined to be true. We get consistent p ↔ q table. //vacuous 空虛的,愚蠢的 ACaRA01.pdf Above is P→Q,1 which is correct choice. <a name="a056_059"> Below is P→Q,4 which is wrong Since red T are defined, we can try other choice. Next is blue T, see whether we get consistent p ↔ q table. 2012-11-27-21-49 here <a name="a056_060"> P→Q,4 is wrong Table 13 //a111272150
pq  pq
TT   T 
TF   F 
FT   F 
FF   F 
pq  pq
TT   T 
TF   F 
FT   F 
FF   F 
<a name="a056_061"> Above table blue F are defined value, this definition is NOT acceptable. Because p → q and p ← q have same result true/false (bold) But p → q and p ← q are opposite operation. <a name="a056_062"> P→Q,4 has two rejects. reject 11:p → q and p ← q same answer. reject 12:p → q and p ∧ q same answer. 2012-11-27-22-12 here Above is P→Q,4 which is wrong <a name="a056_063"> Below is P→Q,3 which is wrong Next see different definition. Purple T/F P→Q,3 is wrong Table 14 //a111272204
pq  pq
TT   T 
TF   F 
FT   F 
FF   T 
pq  pq
TT   T 
TF   F 
FT   F 
FF   T 
<a name="a056_064"> Reject Purple T/F immediately. Same reason as blue. p → q and p ← q have same result true/false (bold) But p → q and p ← q are opposite operation. <a name="a056_065"> P→Q,3 has two rejects. reject 21:p → q and p ← q same answer. reject 22:p → q and p ↔ q same answer. 2012-11-27-22-20 here Above is P→Q,3 which is wrong. <a name="a056_066"> Below is P→Q,2 which is wrong. Below P→Q,2 is important argument. Next see last possible definition. <a name="a056_067"> grey T/F is P→Q,2 Table 15 //a111272221
pq  pq
TT   T 
TF   F 
FT   T 
FF   F 
pq  pq
TT   T 
TF   T 
FT   F 
FF   F 
<a name="a056_068"> This grey table has different p → q and p ← q true/false result Table 16 //a111272224
pq  pq
TT   T 
TF   F 
FT   T 
FF   F 
pq  pq
TT   T 
TF   T 
FT   F 
FF   F 
a202181624 note: tute0056_v04_done_SOLID_REASON.htm first show up table 15 and table 16 It was identical, therefore one of table 15 and table 16 is redundant. <a name="a056_069"> Apply m ∧ n to p → q and p ← q get p ↔ q Table 17 //a111272226
mn  mn
TT   T 
TF   F 
FT   F 
FF   F 
pq  pq
TT   T 
TF   F 
FT   F 
FF   F 
<a name="a056_070"> Last F is NOT acceptable. Above line is important reason. We insist false∧false be false, we get it at m ∧ n column last entry. We insist false↔false be true, but P→Q,2 give us false, see p ↔ q column last entry F. This F reject P→Q,2 . SOLID REASON. <a name="a056_071"> Here m is (p → q) and n is (p ← q) Four grey T/F can not give us consistent logic result. Must reject four grey T/F. <a name="a056_072"> P→Q,2 has two rejects. reject 31:p → q get F↔F=F wrong! (F↔F=T OK) reject 32:p → q and q same logic value. Only first red P→Q,1 T/F give us no contradiction result. 2012-11-27-22-36 stop <a name="a056_073"> 2012-11-27 find SOLID REASON to reject P→Q,2 2012-11-29 find easy reason to reject P→Q,2 because P→Q,2 compound result is same as Q T/F value, P is ignored. This is not acceptable. <a name="a056_074"> Only P→Q,1 give us no contradiction result. Because P→Q,1 is the only correct expression, no second choice. Above word "four possible way" and "defined to be true" are improper. No one say "define" 1+2=3, because 1+2 ONLY =3, no choice. 2012-11-29-23-30 stop <a name="a056_075"> 2012-12-01-09-00 start update 2012-12-01 (tangle∨(~tangle)) → VacuousTruth Next explain the central puzzle "if false then true" is true? why? pay attention to three yellow cells Logic Binary True Table. 理則二元真表 Table 18 //a112010902
PQ  dulP∨Qfulgul P→QhulP↔QP∧Q → can be defined by red column
TT  TTTT TTTT → can NOT be defined by blue col
T F  TT T T F FFF T→F lie be True? absolute reject
F T  TTFF TTFF F→T be T , Vacuous Truth, OK
FF  TFTF TFTF
<a name="a056_076"> "True Table" is different from "Truth Table" "True Table" mean 8 operator P=T,Q=T;P*Q=T list. "Truth Table"=1 equation P=T/F,Q=T/F;P*Q=T/F list. "Total Table"=16 operator P=T/F,Q=T/F;P*Q=T/F list. Let us exam above True Table. dul column all be true, it is dull. forget "P dul Q" operation. P∨Q column fit P OR Q perfect. gul column has compound T/F=P T/F, reject. hul column has compound T/F=Q T/F, reject. P↔Q column fit P iff Q perfect. P∧Q column fit P AND Q perfect. <a name="a056_077"> Only blue and red column left for P→Q to choose. Blue ful column has "if P=true then Q=false" is P→Q=true A true assumption get a false result this compound statement is true? NO! Absolute reject!! Eight possible columns reject/occupied seven columns. Only red column left. <a name="a056_078"> "if false then true" is true? why? Answer is //vacuously true,ACaRA01.pdf Red column is the only seat left, Mr. P→Q, please take it, please tolerate. and "if P=false then Q=true" be true which is not as bad as "if P=true then Q=false" be true 2012-12-01-09-26 stop <a name="a056_079"> 2012-12-01-10-55 start P→Q has "vacuously true", LiuHH try to find different definition for P→Q. Only candidate is ful column. Table 19 has 4 columns total 16 T/F, next table indicate comments involve which T/F (involve bold T/F). table 18 Table 19 ful,P→Q major table //a112011056
PQ  fulrow 3/4 involve bold T/F, ignore thin grey F. P→Q
TT  TP→Q say reject next blue line, reject ful column T
TF  T"if P=true then Q=false" be true absolute NO! F
FT  F"if P=false then Q=true" be true vacuous true T
FF  TAbove red not as bad as blue. tolerate bold red T
Reject ful column as P→Q definition, Accept "P→Q" column as P→Q definition. 2012-12-01-11-31 stop //minor table <a name="a056_080"> 2012-12-01-20-50 target at "vacuous" (open local saved copy, below same) http://www.uow.edu.au/~bmaloney/wuct121/LogicTeacher.pdf LogicTeacher.pdf no "vacuous" 2012-12-01-20-55 open http://schmidt.ucg.ie/ma184/screen-4x4.pdf ma184_screen-4x4.pdf very hard to read. <a name="a056_081"> 2012-12-01-21-15 open http://www.abstractmath.org/MM/MMConditional.htm has more about Vacuous truth [[ <a name="a056_082"> Vacuous truth The last two lines of the truth table for conditional assertions mean that if the hypothesis of the assertion is false, then the assertion is automatically true. In the case that "If P then Q" is true because P is false, the assertion is said to be vacuously true. <a name="a056_083"> The word "vacuous" refers to the fact that in that case the conditional assertion says nothing interesting about either the hypothesis or the conclusion. In particular, the conditional assertion may be true even if the conclusion is false (because of the last line of the truth table). ]] //above copied from MMConditional.htm <a name="a056_084"> 2012-12-01-21-22 open http://mysite.verizon.net/thelogos/Logic_Notes01.pdf not mention "Vacuous truth" 2012-12-01-21-26 open http://www.ksmodern1.com/Logic%20part%203.pdf not mention "Vacuous truth" <a name="a056_085"> 2012-12-01-21-28 open http://www.math.umn.edu/~jodeit/course/ACaRA01.pdf LiuHH first "Vacuous truth" reading 2012-12-01-21-30 open http://faculty.swosu.edu/michael.dougherty/book/chapter01.pdf "Vacuous truth" page 6/72 to 12/72 page 29/72 to 39/72, page 49/72, page 62/72, page 65/72 2012-12-01-21-35 stop <a name="a056_086"> 2012-12-02-15-05 start update 2012-12-02 2012-12-01-17-35 upload 90,968 tute0056.htm 2012-12-01-18-07 upload 78,986 tute0056.htm 2012-12-01-18-07 erased [a name="a056_075"] to [a name="a056_079"] Because Liu,Hsinhan suspect "075" to "079" may be incomplete or incorrect. <a name="a056_087"> 2012-12-01-19-?? LiuHH re-read Table 19 carefully and find Table 19 is OK. To explain "if P=false then Q=true" be true Table 19 is major corner. Next, Table 20 is minor corner. <a name="a056_088"> Table 20 ful,P→Q minor table //a112021514
PQ  fulrow 3/4 involve bold T/F, ignore thin grey T. P→Q
TT  TTrue assumption get false result, a lie statement. T
T F  T "if P=true then Q=false" be false correct P→Q F
F T  F "if P=false then Q=true" be false naive guess T
FF  Tnaive argue F→T be F, open T→F be T, ERROR! T
<a name="a056_088key"> //major table If ask F→T be F (wrong), ful column row 3 has F T F it satisfy this requirement, then take ful column as P→Q definition column (wrong) The fatal result is that must accept "T→F be T", since ful column row two has T F T. BUT admit "T→F be T" which is absolute wrong!! We must refuse take ful column as P→Q definition. Now only P→Q column can be P→Q definition. // a202171118 done [a056_088key] <a name="a056_089"> Why naive argue F→T be F, open T→F be T, ERROR! Because P→Q definition has only two columns to choose. Please see table 18 and a056_076 and a056_077 If insist F→T be F (wrong) then must choose blue ful column as P→Q definition <a name="a056_090"> In blue ful column, it has (open a way) T→F be T A true assumption get a false result, this compound statement is a lie, a lie be true? NO. We must reject blue ful column as P→Q definition Only red column left as P→Q definition. <a name="a056_091"> Truth Table T/F should not be altered. P/Q column change T/F value that has no meaning. Because P/Q column has elementary T/F, they are a complete combination. <a name="a056_092"> Compound column (for example P→Q col.) change T/F value that has no meaning either. Because refer to Total Table, change T/F value in compound column, that is just choose another column. But every column are well defined or well rejected. 2012-12-02-15-58 stop <a name="a056_093"> update 2012-12-04 2012-12-04-09-00 start ltth() lttt() Logic Binary Total Table may appear in document multiple times. Write function ltth() and function lttt() display as following. ltth() is html table code without Chinese. lttt() is text table code without Chinese. ltth(1) is html table code with Chinese. lttt(1) is text table code with Chinese. <a name="a056_094"> If you use next eight red code lines, you must include function ltth() //LogicTotalTableHtml and function lttt() //LogicTotalTableText and function ltrh() //LogicTRuetableHtml and function ltrt() //LogicTRuetableText code body in your web page. TotalTable and TrueTable last row is column name. <a name="a056_095"> Code <script language="javascript">ltth(1)</script> get <a name="a056_096"> Next table in side of <pre></pre> too wide. Code <script language="javascript">lttt(1)</script> get
<a name="a056_097"> Next table out side of <pre></pre>
Code <script language="javascript">ltth()</script> get

<a name="a056_098"> Next table out side of <pre></pre>
Code <script language="javascript">lttt()</script> get
2012-12-04-09-57 here ltth() lttt() 
<a name="a056_099">
Code <script language="javascript">ltrh(1)</script> get 


<a name="a056_100">
Code <script language="javascript">ltrh()</script> get 


<a name="a056_101">
Code <script language="javascript">ltrt(1)</script> get 


<a name="a056_102">
Code <script language="javascript">ltrt()</script> get 

2012-12-04-10-15 here ltrh() ltrt()

<a name="a056_103">
If you want to display 
Logic Binary False Table
LiuHH did not code it. Please use 
Logic Binary Total Table
since its right half is False Table
False Table include next columns 
vul xor tul sul rul qul pul nul 
2012-12-04-10-23 stop 

<a name="a056_104">
2012-12-23-19-38 start 
update 2012-12-23 add link to 
http://freeman2.com/logictt2.htm
Math Logic Truth Table Generator.
logictt2.htm is a tool for tute0056.htm
When Liu,Hsinhan study Math Logic pages 
LiuHH need convert logic equation to 
a Logic Truth Table. Do it by hand? 
<a name="a056_105">
One or two may be. But for future long 
time consideration, it is better write 
a program to convert logic equation to 
a truth table. Finally, logictt2.htm 
is here. logictt2.htm Box03 output is 
my major fruit. Please visit 
http://freeman2.com/logictt2.htm
http://freeman2.com/logictt1.htm Chinese 
hope you find it is useful.
2012-12-23-19-45 stop

<a name="a056_106">
2013-02-18-12-45 start 
update 2013-02-18 make minor change in 
study notes explanation, make concept  
expression more clear, for example 
a056_088key When write Chinese version, 
made such change. 
2013-02-18-12-47 stop 

<a name="a056_801">
2012-11-30-11-24 done first proofread
2012-12-01-12-02 done second proofread
2013-02-18-18-17 done third proofread

~ ∨ ∧ → ← ↔ ⇐ ⇒ ⇔








Javascript index
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Save graph code to same folder as htm files.
http://freeman2.com/jsgraph2.js   local


file name tute0056.htm mean
TUTor, English, 56 th .htm
Chinese version is tutc0056.htm


http://freeman2.com/tute0056.htm
First upload 2012-11-30

Thank you for visiting Freeman's page.
Freeman Liu,Hsinhan 劉鑫漢
2012-11-30-11-27

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