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Let-A-B-and-C-are-three-statments-A-B-C-A-C-B-A-B-C-A-B-A-Prove-or-disprove-




Question Number 1656 by Rasheed Soomro last updated on 29/Aug/15
Let A,B and C are three statments.   (A⇒B⇒C⇒A) ⇒^(?) (C⇒B)   (A⇒B⇒C⇒A) ⇒^(?) (B⇒A)   Prove or disprove.
LetA,BandCarethreestatments.(ABCA)?(CB)(ABCA)?(BA)Proveordisprove.
Commented by Yozzian last updated on 29/Aug/15
The best approach to showing  whether or not those statements  are tautologies or contradictions  is by use of truth tables.
Thebestapproachtoshowingwhetherornotthosestatementsaretautologiesorcontradictionsisbyuseoftruthtables.
Commented by 112358 last updated on 29/Aug/15
A⇒B⇒C⇒A≡^? ((A⇒B)⇒C)⇒A
ABCA?((AB)C)A
Commented by Yozzy last updated on 29/Aug/15
Let p=(A⇒B⇒C⇒A)⇒(C⇒B)            r=(A⇒B⇒C⇒A)⇒(B⇒A)  I′m assuming that A⇒B⇒C⇒A≡((A⇒B)⇒C)⇒A≡(A⇒B)⇒(C⇒A) .  Let q=(A⇒B)⇒(C⇒A)              1   2   3       4           5                               6                              7         8     9        10      11    12 (column numbers)  A∣B∣C∣A⇒B∣(A⇒B)⇒C∣((A⇒B)⇒C)⇒A∣ C⇒B∣ p∣C⇒A∣q  ∣B⇒A∣ r  0∣  0 ∣0        1               0                             1                             1      1       1       1        1       1  0∣  0 ∣1        1               1                             0                             0      1       0       0        1       1  0∣  1 ∣0        1               0                             1                             1      1       1       1        0       0  0∣  1 ∣1        1               1                             0                             1      1       0       0        0       1  1∣  0 ∣0        0               1                             1                             1      1       1       1        1       1  1∣  0 ∣1        0               1                             1                             0      0       1       1        1       1  1∣  1 ∣0        1               0                             1                             1      1       1       1        1       1  1∣  1 ∣1        1               1                             1                             1      1       1       1        1       1  (1 indicates a statement is true while 0 indicates a statement is false.)  From the above truth table column 8, we have that p=(A⇒B⇒C⇒A)⇒(C⇒B)≡1 if we  have any logic combination of A,B and C other than A=1,B=0,C=1.  By Boolean algebra,                p=(1⇒0⇒1⇒1)⇒(1⇒0)                 =(0⇒1⇒1)⇒0                 =(1⇒1)⇒0                 =∽(1⇒1)∨0                 =∽1∨0                 =0∨0              p=0  (A⇒B⇒C⇒⇒A)⇏(C⇒B) if A is true,B is false and C is true.  Also, column 12 shows that (A⇒B⇒C⇒A)⇒(B⇒A) is false iff A=0 ,  B=1 and C=0.
Letp=(ABCA)(CB)r=(ABCA)(BA)ImassumingthatABCA((AB)C)A(AB)(CA).Letq=(AB)(CA)123456789101112(columnnumbers)ABCAB(AB)C((AB)C)ACBpCAqBAr000101111111001110010011010101111100011110110001100011111111101011001111110101111111111111111111(1indicatesastatementistruewhile0indicatesastatementisfalse.)Fromtheabovetruthtablecolumn8,wehavethatp=(ABCA)(CB)1ifwehaveanylogiccombinationofA,BandCotherthanA=1,B=0,C=1.ByBooleanalgebra,p=(1011)(10)=(011)0=(11)0=∽(11)0=∽10=00p=0(ABC⇒⇒A)(CB)ifAistrue,BisfalseandCistrue.Also,column12showsthat(ABCA)(BA)isfalseiffA=0,B=1andC=0.
Commented by Rasheed Soomro last updated on 30/Aug/15
Thanks for so much labour.  I have not been much  involved in logic.   So  if you see no meaning in my questions please excuse  me.   Is there a transitive law of implication?  A⇒B ,B⇒C then A⇒C ?  ′If A is true then B is true′ ,′ If B is true then C is true′                  ⇒   ′If A is true then C is true′   ?  Can we write A⇒B⇒C⇒A in place of  A⇒B, B⇒C,C⇒A  ?
Thanksforsomuchlabour.Ihavenotbeenmuchinvolvedinlogic.Soifyouseenomeaninginmyquestionspleaseexcuseme.Isthereatransitivelawofimplication?AB,BCthenAC?IfAistruethenBistrue,IfBistruethenCistrueIfAistruethenCistrue?CanwewriteABCAinplaceofAB,BC,CA?
Commented by Yozzy last updated on 30/Aug/15
Propose that p=A⇒B,q=B⇒C and r=A⇒C .  To prove there being a transitive property of implication is like saying (I think) to   show that [(A⇒B)∧(B⇒C)]⇒(A⇒C)=T, i.e (p∧q)⇒r  is a tautology.  Without truth tables, I′ll attempt to use the laws of Boolean algebra  to simplify the logic expression to hopefully end up having a tautology.  Note that μ⇒z≡∽μ∨z.  [(A⇒B)∧(B⇒C)]⇒(A⇒C)≡[(∽A∨B)∧(∽B∨C)]⇒(∽A∨C)  lhs≡∽[(∽A∨B)∧(∽B∨C)]∨(∽A∨C)  lhs≡[∽(∽A∨B)∨∽(∽B∨C)]∨(∽A∨C)                  De Morgan′s law  lhs≡[(A∧∽B)∨(B∧∽C)]∨(∽A∨C)                          De Morgan′s law, Double negation law  lhs≡[∽A∨(A∧∽B)]∨[C∨(B∧∽C)]                            Commutative law,Associative law  lhs≡[(∽A∨A)∧(∽A∨∽B)]∨[(C∨B)∧(C∨∽C)]    Distribution law  lhs≡[T∧(∽A∨∽B)]∨[(C∨B)∧T]                                      Complementary law  lhs≡(∽A∨∽B)∨(C∨B)                                                         Identity law  lhs≡(∽B∨B)∨(C∨∽A)                                                          Commutative law,Associative law  lhs≡T∨(C∨∽A)                                                                       Complementary law  lhs≡T                                                                                             Domination law   The simplification of the Boolean statement has given us a constant  truth value of T, so the statement is perpetually true. Hence,a transitive  property of the implication logic as been proven. ■
Proposethatp=AB,q=BCandr=AC.Toprovetherebeingatransitivepropertyofimplicationislikesaying(Ithink)toshowthat[(AB)(BC)](AC)=T,i.e(pq)risatautology.Withouttruthtables,IllattempttousethelawsofBooleanalgebratosimplifythelogicexpressiontohopefullyenduphavingatautology.Notethatμz≡∽μz.[(AB)(BC)](AC)[(AB)(BC)](AC)lhs≡∽[(AB)(BC)](AC)lhs[(AB)(BC)](AC)DeMorganslawlhs[(AB)(BC)](AC)DeMorganslaw,Doublenegationlawlhs[A(AB)][C(BC)]Commutativelaw,Associativelawlhs[(AA)(AB)][(CB)(CC)]Distributionlawlhs[T(AB)][(CB)T]Complementarylawlhs(AB)(CB)Identitylawlhs(BB)(CA)Commutativelaw,AssociativelawlhsT(CA)ComplementarylawlhsTDominationlawThesimplificationoftheBooleanstatementhasgivenusaconstanttruthvalueofT,sothestatementisperpetuallytrue.Hence,atransitivepropertyoftheimplicationlogicasbeenproven.◼
Commented by Yozzy last updated on 30/Aug/15
A∣B∣C∣A⇒B∣B⇒C∣A⇒C∣(A⇒B)∧(B⇒C)∣[(A⇒B)∧(B⇒C)]⇒(A⇒C)  0    0  0     1              1           1                        1                                  1  0    0  1     1              1           1                        1                                  1  0    1  0     1              0           1                        0                                  1  0    1  1     1              1           1                        1                                  1  1    0  0     0              1           0                        0                                  1  1    0  1     0              1           1                        0                                  1  1    1  0     1              0           0                        0                                  1  1    1  1     1              1           1                        1                                  1  The last column to the right indicates that the statement  [(A⇒B)∧(B⇒C)]⇒(A⇒C) is always true. Therefore, this proves  the existance of a transitive property of the implication logic.  (I′ll check over my Boolean simplification.Checked.)
ABCABBCAC(AB)(BC)[(AB)(BC)](AC)0001111100111111010101010111111110001001101011011101000111111111Thelastcolumntotherightindicatesthatthestatement[(AB)(BC)](AC)isalwaystrue.Therefore,thisprovestheexistanceofatransitivepropertyoftheimplicationlogic.(IllcheckovermyBooleansimplification.Checked.)
Commented by Rasheed Soomro last updated on 30/Aug/15
THANKSSsssssss!  Yozzian=^(?) Yozzy
THANKSSsssssss!Yozzian=?Yozzy
Commented by Yozzy last updated on 30/Aug/15
I use my cell phones (2) and my tablet. I′ve been the only one commenting  on this question. I used my tablet to answer the question because of the  long truth table.
Iusemycellphones(2)andmytablet.Ivebeentheonlyonecommentingonthisquestion.Iusedmytablettoanswerthequestionbecauseofthelongtruthtable.
Commented by Rasheed Soomro last updated on 30/Aug/15
I have also more than one device and I use two ID′s:  Rasheed Soomro     and        Rasheed Ahmad
IhavealsomorethanonedeviceandIusetwoIDs:RasheedSoomroandRasheedAhmad
Commented by 112358 last updated on 30/Aug/15
I see.
Isee.
Commented by 123456 last updated on 30/Aug/15
XD 2^n_i
XD2ni
Commented by Rasheed Ahmad last updated on 30/Aug/15
XD 2^n_i   ???
XD2ni???

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