What are we doing when we are proofing a theorem “if A, then B”?

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What are we doing when we are proofing a theorem "if A, then B"?Are we proving that "if A, then B" is true? We know "if A, then B" is true when A is true and B is true. So are we proving that B is try when A is true ? But we also know "if A, then B" is true when A is false and B is true ,or A is false and B is false. So can we prove "if A, then B" by proving B is false when A is false ? Or B is true when A is false ?




logic proof-writing




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  • 2




    math.stackexchange.com/questions/71809/… this thread has a good explanation imo
    – Sar
    Jul 24 at 4:12














up vote
1
down vote

favorite












What are we doing when we are proofing a theorem "if A, then B"?Are we proving that "if A, then B" is true? We know "if A, then B" is true when A is true and B is true. So are we proving that B is try when A is true ? But we also know "if A, then B" is true when A is false and B is true ,or A is false and B is false. So can we prove "if A, then B" by proving B is false when A is false ? Or B is true when A is false ?




logic proof-writing




share|cite|improve this question















  • 2




    math.stackexchange.com/questions/71809/… this thread has a good explanation imo
    – Sar
    Jul 24 at 4:12












up vote
1
down vote

favorite









up vote
1
down vote

favorite











What are we doing when we are proofing a theorem "if A, then B"?Are we proving that "if A, then B" is true? We know "if A, then B" is true when A is true and B is true. So are we proving that B is try when A is true ? But we also know "if A, then B" is true when A is false and B is true ,or A is false and B is false. So can we prove "if A, then B" by proving B is false when A is false ? Or B is true when A is false ?




logic proof-writing




share|cite|improve this question











What are we doing when we are proofing a theorem "if A, then B"?Are we proving that "if A, then B" is true? We know "if A, then B" is true when A is true and B is true. So are we proving that B is try when A is true ? But we also know "if A, then B" is true when A is false and B is true ,or A is false and B is false. So can we prove "if A, then B" by proving B is false when A is false ? Or B is true when A is false ?





logic proof-writing





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share|cite|improve this question




share|cite|improve this question









asked Jul 24 at 4:07









user579146

61




61







  • 2




    math.stackexchange.com/questions/71809/… this thread has a good explanation imo
    – Sar
    Jul 24 at 4:12












  • 2




    math.stackexchange.com/questions/71809/… this thread has a good explanation imo
    – Sar
    Jul 24 at 4:12







2




2




math.stackexchange.com/questions/71809/… this thread has a good explanation imo
– Sar
Jul 24 at 4:12




math.stackexchange.com/questions/71809/… this thread has a good explanation imo
– Sar
Jul 24 at 4:12










1 Answer
1






active

oldest

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up vote
0
down vote













We are show that the statement



$$A to B$$



is a tautology.



In other words, if $A$ is true, than there is no such a case where $B$ is false (provided $A$ is true)






share|cite|improve this answer





















  • But What does it mean when we say "if A then B" is true ?
    – user579146
    Jul 24 at 4:27










  • Say, for example, $A:$ This is a sunny day, and $B:$ Today is going to be a good day, and you have somehow proved that $Ato B$. Then if you wake up a morning and see that that day is a sunny day, then you are %$100$ sure that that day is going to be a good day.
    – onurcanbektas
    Jul 24 at 4:29











  • In other words, provided that the statement $A$ is true, then the statement $B$ is also true.
    – onurcanbektas
    Jul 24 at 4:30






  • 1




    So can I say proving "if A, then B" is to show that "if A, then B" is true ?
    – user579146
    Jul 24 at 4:36










  • @user579146 Yes that is correct. Actually, that is what a tautology means. It means the statement $(A to B)$ is always true regardless of the values of $A$.
    – onurcanbektas
    Jul 24 at 4:37











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote













We are show that the statement



$$A to B$$



is a tautology.



In other words, if $A$ is true, than there is no such a case where $B$ is false (provided $A$ is true)






share|cite|improve this answer





















  • But What does it mean when we say "if A then B" is true ?
    – user579146
    Jul 24 at 4:27










  • Say, for example, $A:$ This is a sunny day, and $B:$ Today is going to be a good day, and you have somehow proved that $Ato B$. Then if you wake up a morning and see that that day is a sunny day, then you are %$100$ sure that that day is going to be a good day.
    – onurcanbektas
    Jul 24 at 4:29











  • In other words, provided that the statement $A$ is true, then the statement $B$ is also true.
    – onurcanbektas
    Jul 24 at 4:30






  • 1




    So can I say proving "if A, then B" is to show that "if A, then B" is true ?
    – user579146
    Jul 24 at 4:36










  • @user579146 Yes that is correct. Actually, that is what a tautology means. It means the statement $(A to B)$ is always true regardless of the values of $A$.
    – onurcanbektas
    Jul 24 at 4:37















up vote
0
down vote













We are show that the statement



$$A to B$$



is a tautology.



In other words, if $A$ is true, than there is no such a case where $B$ is false (provided $A$ is true)






share|cite|improve this answer





















  • But What does it mean when we say "if A then B" is true ?
    – user579146
    Jul 24 at 4:27










  • Say, for example, $A:$ This is a sunny day, and $B:$ Today is going to be a good day, and you have somehow proved that $Ato B$. Then if you wake up a morning and see that that day is a sunny day, then you are %$100$ sure that that day is going to be a good day.
    – onurcanbektas
    Jul 24 at 4:29











  • In other words, provided that the statement $A$ is true, then the statement $B$ is also true.
    – onurcanbektas
    Jul 24 at 4:30






  • 1




    So can I say proving "if A, then B" is to show that "if A, then B" is true ?
    – user579146
    Jul 24 at 4:36










  • @user579146 Yes that is correct. Actually, that is what a tautology means. It means the statement $(A to B)$ is always true regardless of the values of $A$.
    – onurcanbektas
    Jul 24 at 4:37













up vote
0
down vote










up vote
0
down vote









We are show that the statement



$$A to B$$



is a tautology.



In other words, if $A$ is true, than there is no such a case where $B$ is false (provided $A$ is true)






share|cite|improve this answer













We are show that the statement



$$A to B$$



is a tautology.



In other words, if $A$ is true, than there is no such a case where $B$ is false (provided $A$ is true)







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 24 at 4:10









onurcanbektas

3,0141832




3,0141832











  • But What does it mean when we say "if A then B" is true ?
    – user579146
    Jul 24 at 4:27










  • Say, for example, $A:$ This is a sunny day, and $B:$ Today is going to be a good day, and you have somehow proved that $Ato B$. Then if you wake up a morning and see that that day is a sunny day, then you are %$100$ sure that that day is going to be a good day.
    – onurcanbektas
    Jul 24 at 4:29











  • In other words, provided that the statement $A$ is true, then the statement $B$ is also true.
    – onurcanbektas
    Jul 24 at 4:30






  • 1




    So can I say proving "if A, then B" is to show that "if A, then B" is true ?
    – user579146
    Jul 24 at 4:36










  • @user579146 Yes that is correct. Actually, that is what a tautology means. It means the statement $(A to B)$ is always true regardless of the values of $A$.
    – onurcanbektas
    Jul 24 at 4:37

















  • But What does it mean when we say "if A then B" is true ?
    – user579146
    Jul 24 at 4:27










  • Say, for example, $A:$ This is a sunny day, and $B:$ Today is going to be a good day, and you have somehow proved that $Ato B$. Then if you wake up a morning and see that that day is a sunny day, then you are %$100$ sure that that day is going to be a good day.
    – onurcanbektas
    Jul 24 at 4:29











  • In other words, provided that the statement $A$ is true, then the statement $B$ is also true.
    – onurcanbektas
    Jul 24 at 4:30






  • 1




    So can I say proving "if A, then B" is to show that "if A, then B" is true ?
    – user579146
    Jul 24 at 4:36










  • @user579146 Yes that is correct. Actually, that is what a tautology means. It means the statement $(A to B)$ is always true regardless of the values of $A$.
    – onurcanbektas
    Jul 24 at 4:37
















But What does it mean when we say "if A then B" is true ?
– user579146
Jul 24 at 4:27




But What does it mean when we say "if A then B" is true ?
– user579146
Jul 24 at 4:27












Say, for example, $A:$ This is a sunny day, and $B:$ Today is going to be a good day, and you have somehow proved that $Ato B$. Then if you wake up a morning and see that that day is a sunny day, then you are %$100$ sure that that day is going to be a good day.
– onurcanbektas
Jul 24 at 4:29





Say, for example, $A:$ This is a sunny day, and $B:$ Today is going to be a good day, and you have somehow proved that $Ato B$. Then if you wake up a morning and see that that day is a sunny day, then you are %$100$ sure that that day is going to be a good day.
– onurcanbektas
Jul 24 at 4:29













In other words, provided that the statement $A$ is true, then the statement $B$ is also true.
– onurcanbektas
Jul 24 at 4:30




In other words, provided that the statement $A$ is true, then the statement $B$ is also true.
– onurcanbektas
Jul 24 at 4:30




1




1




So can I say proving "if A, then B" is to show that "if A, then B" is true ?
– user579146
Jul 24 at 4:36




So can I say proving "if A, then B" is to show that "if A, then B" is true ?
– user579146
Jul 24 at 4:36












@user579146 Yes that is correct. Actually, that is what a tautology means. It means the statement $(A to B)$ is always true regardless of the values of $A$.
– onurcanbektas
Jul 24 at 4:37





@user579146 Yes that is correct. Actually, that is what a tautology means. It means the statement $(A to B)$ is always true regardless of the values of $A$.
– onurcanbektas
Jul 24 at 4:37













 

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