Most presentations of the axioms of group theory seem to giveboth right and left identities and inverses, which turns out to be
https://dcproof.com/GroupLeftRightIdentityInverses.htm (only 123 lines)
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Most presentations of the axioms of group theory seem to give both right and left identities and inverses, which turns out to be somewhat redundant. Here, given the axioms for group (g,*) with only right identity and inverses, we formally prove thatthe right identity and inverses are also a left identity and inverse respectively.
https://dcproof.com/GroupLeftRightIdentityInverses.htm (only 123 lines)
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
On Wednesday, July 5, 2023 at 10:07:53 PM UTC+3, Dan Christensen wrote:the right identity and inverses are also a left identity and inverse respectively.
Most presentations of the axioms of group theory seem to give both right and left identities and inverses, which turns out to be somewhat redundant. Here, given the axioms for group (g,*) with only right identity and inverses, we formally prove that
https://dcproof.com/GroupLeftRightIdentityInverses.htm (only 123 lines)
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.comWhy people are attacking you constantly Dan C?
Visit my Math Blog at http://www.dcproof.wordpress.com
Doesn't that suggest something to you where you don't think of? No wonder!
But certainly you are suffering constantly from something in mind 🙃! Sure
BKK
Why don't you show a proof BKK, make yourself usefulthat the right identity and inverses are also a left identity and inverse respectively.
once in your life. So far you didn't post anything interesting.
bassam karzeddin schrieb am Donnerstag, 14. September 2023 um 20:32:33 UTC+2:
On Wednesday, July 5, 2023 at 10:07:53 PM UTC+3, Dan Christensen wrote:
Most presentations of the axioms of group theory seem to give both right and left identities and inverses, which turns out to be somewhat redundant. Here, given the axioms for group (g,*) with only right identity and inverses, we formally prove
https://dcproof.com/GroupLeftRightIdentityInverses.htm (only 123 lines)
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.comWhy people are attacking you constantly Dan C?
Visit my Math Blog at http://www.dcproof.wordpress.com
Doesn't that suggest something to you where you don't think of? No wonder!
But certainly you are suffering constantly from something in mind 🙃! Sure
BKK
Come on Dan Christensen. Do you really want to tell me:
- You CANNOT show that U is redundant in Trichotomy,
representable by T and F?
On Thursday, September 14, 2023 at 2:26:41 PM UTC-4, Mild Shock wrote:
Come on Dan Christensen. Do you really want to tell me:
- You CANNOT show that U is redundant in Trichotomy,
representable by T and F?
[snip]
Apparently, you cannot tell a trichotomy from a dichotomy. See my posting just now in the thread "The Liar Paradox: My latest blog posting" at sci.logic.
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
The proof is as easy as follows:
- We don't need to use predicates
- We don't need to use sets
- We can show it with propositional variables
All we need to prove is:
Equivalence: (((¬T ∧ ¬F) ↔ U)
Trichotomy II: ¬(T∧F)
Trichotomy I: ((T ∨ (F∨U)) ∧ (¬(T∧F) ∧ (¬(T∧U) ∧ ¬(F∧U))))
Equivalence & Trichotomy II <=> Trichotomy I
Here is a proof:
(((¬T ∧ ¬F) ↔ U) ∧ ¬(T∧F)) ↔ ((T ∨ (F∨U)) ∧ (¬(T∧F) ∧ (¬(T∧U) ∧ ¬(F∧U)))) is valid.
https://www.umsu.de/trees/#(~3T~1~3F~4U)~1~3(T~1F)~4(T~2F~2U)~1~3(T~1F)~1~3(T~1U)~1~3(F~1U)
Easy, wasn't it?
Dan Christensen schrieb am Donnerstag, 14. September 2023 um 22:14:31 UTC+2:
On Thursday, September 14, 2023 at 2:26:41 PM UTC-4, Mild Shock wrote:
Come on Dan Christensen. Do you really want to tell me:
- You CANNOT show that U is redundant in Trichotomy,
representable by T and F?
[snip]
Apparently, you cannot tell a trichotomy from a dichotomy. See my posting just now in the thread "The Liar Paradox: My latest blog posting" at sci.logic.
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Or instead of using Wolfgang Schwarz tree tool, you
can also look at truth tables, and see that it is the same:
Equivalence & Trichotomy II
Fs Ts Us (((¬Ts ∧ ¬Fs) ↔ Us) ∧ ¬(Ts ∧ Fs))
F F T T
F T F T
T F F T
https://web.stanford.edu/class/cs103/tools/truth-table-tool/
Trichotomy I
Fs Ts Us ((Ts ∨ (Fs ∨ Us)) ∧ (¬(Ts ∧ Fs) ∧ (¬(Ts ∧ Us) ∧ ¬(Fs ∧ Us))))
F F T T
F T F T
T F F T
https://web.stanford.edu/class/cs103/tools/truth-table-tool/
Q.E.D.
Mild Shock schrieb am Donnerstag, 14. September 2023 um 22:26:03 UTC+2:
The proof is as easy as follows:
- We don't need to use predicates
- We don't need to use sets
- We can show it with propositional variables
All we need to prove is:
Equivalence: (((¬T ∧ ¬F) ↔ U)
Trichotomy II: ¬(T∧F)
Trichotomy I: ((T ∨ (F∨U)) ∧ (¬(T∧F) ∧ (¬(T∧U) ∧ ¬(F∧U))))
Equivalence & Trichotomy II <=> Trichotomy I
Here is a proof:
(((¬T ∧ ¬F) ↔ U) ∧ ¬(T∧F)) ↔ ((T ∨ (F∨U)) ∧ (¬(T∧F) ∧ (¬(T∧U) ∧ ¬(F∧U)))) is valid.
https://www.umsu.de/trees/#(~3T~1~3F~4U)~1~3(T~1F)~4(T~2F~2U)~1~3(T~1F)~1~3(T~1U)~1~3(F~1U)
Easy, wasn't it?
Dan Christensen schrieb am Donnerstag, 14. September 2023 um 22:14:31 UTC+2:
On Thursday, September 14, 2023 at 2:26:41 PM UTC-4, Mild Shock wrote:
Come on Dan Christensen. Do you really want to tell me:
- You CANNOT show that U is redundant in Trichotomy,
representable by T and F?
[snip]
Apparently, you cannot tell a trichotomy from a dichotomy. See my posting just now in the thread "The Liar Paradox: My latest blog posting" at sci.logic.
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Most presentations of the axioms of group theory seem to give both right and left identities and inverses, which turns out to be somewhat redundant. Here, given the axioms for group (g,*) with only right identity and inverses, we formally prove thatthe right identity and inverses are also a left identity and inverse respectively.
https://dcproof.com/GroupLeftRightIdentityInverses.htm (only 123 lines)https://proofwiki.org/wiki/Left_Inverse_for_All_is_Right_Inverse
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
The proof is as easy as follows:
On Thursday, September 14, 2023 at 9:39:51 PM UTC+3, Mild Shock wrote:that the right identity and inverses are also a left identity and inverse respectively.
Why don't you show a proof BKK, make yourself useful
once in your life. So far you didn't post anything interesting.
bassam karzeddin schrieb am Donnerstag, 14. September 2023 um 20:32:33 UTC+2:
On Wednesday, July 5, 2023 at 10:07:53 PM UTC+3, Dan Christensen wrote:
Most presentations of the axioms of group theory seem to give both right and left identities and inverses, which turns out to be somewhat redundant. Here, given the axioms for group (g,*) with only right identity and inverses, we formally prove
https://dcproof.com/GroupLeftRightIdentityInverses.htm (only 123 lines)
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.comWhy people are attacking you constantly Dan C?
Visit my Math Blog at http://www.dcproof.wordpress.com
Doesn't that suggest something to you where you don't think of? No wonder!
But certainly you are suffering constantly from something in mind 🙃! Sure
Why can't you Mild Shock version well-understand my too simple proofs instead?BKK
I do believe that I have proven ALL my unique & so peculiar & rarest historical claims with too simple elementary & numerical "irrefutable" proofs FOR SURE
But so unfortunately, since only my unique vision & proofs are against the academic mainstreams dogmatic beliefs & their achievements, then certainly it would be globally denied & fought to death as well
Go learn my proofs & rewrite them in a logical mathematical way you are pretending!
Bassam karzeddin
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