.How do I prove an implication of disjunction?

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I want to prove that
A ⇒ B (1).

I write the implication
A ⇒ (B ∨ C) (2).

I can prove by contradiction that (2) is true, then I prove that C is false.

Is this means that the implication
A ⇒ B is true?

logic

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asked Jul 27 at 14:15

Ilyes

12

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up vote
0
down vote

favorite

I want to prove that
A ⇒ B (1).

I write the implication
A ⇒ (B ∨ C) (2).

I can prove by contradiction that (2) is true, then I prove that C is false.

Is this means that the implication
A ⇒ B is true?

logic

share|cite|improve this question

asked Jul 27 at 14:15

Ilyes

12

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I want to prove that
A ⇒ B (1).

I write the implication
A ⇒ (B ∨ C) (2).

I can prove by contradiction that (2) is true, then I prove that C is false.

Is this means that the implication
A ⇒ B is true?

logic

share|cite|improve this question

asked Jul 27 at 14:15

Ilyes

12

I want to prove that
A ⇒ B (1).

I write the implication
A ⇒ (B ∨ C) (2).

I can prove by contradiction that (2) is true, then I prove that C is false.

Is this means that the implication
A ⇒ B is true?

logic

share|cite|improve this question

asked Jul 27 at 14:15

Ilyes

12

share|cite|improve this question

share|cite|improve this question

asked Jul 27 at 14:15

Ilyes

12

asked Jul 27 at 14:15

Ilyes

12

asked Jul 27 at 14:15

Ilyes

12

12

add a comment | 

add a comment | 

2 Answers
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0
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If $C equiv 0$, then $B lor C equiv B lor 0 equiv$ B, so that $(A implies B) equiv (A implies (B lor C ))$. So if $(A implies (B lor C ))$ takes value $1$, so does $(A implies B)$, and conversely for $0$.

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answered Jul 27 at 14:46

ippiki-ookami

303216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have proved that C is false after proving that (1) is true by contradiction. In the proof of (1), I have used (not C).
  – Ilyes
  Jul 27 at 17:07
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  When C is false both your (1) and (2) are logically equivalent. That means they will take the same truth values, so if you proved that one is true, so should the other be, do you agree ?
  – ippiki-ookami
  Jul 27 at 17:20
  

  
  
  
  

add a comment | 

up vote
0
down vote

Are you asking whether $Ato B$ is a valid inference from $Ato(Bvee C)$ and $neg C$?   It is.

Take $neg C$ and $Ato(Bvee C)$ as premises.   By assuming $A$, you may derive that $B$ or $C$ will be true, but $C$ is promised to be false, leaving only that $B$ may be true.   Therefore $Ato B$ is entailed by the premises.

So if you have proven $Ato (Bvee C)$ and have proven $neg C$, then you may infer that $Ato B$ is provable.

share|cite|improve this answer

answered Jul 27 at 22:18

Graham Kemp

80k43275

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I agree with the two responses. When we can't prove directly (1), The proof of (2) is an indirect proof of (1), if C is false.
  – Ilyes
  Jul 27 at 22:40
  

  
  
  
  

add a comment | 

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

If $C equiv 0$, then $B lor C equiv B lor 0 equiv$ B, so that $(A implies B) equiv (A implies (B lor C ))$. So if $(A implies (B lor C ))$ takes value $1$, so does $(A implies B)$, and conversely for $0$.

share|cite|improve this answer

answered Jul 27 at 14:46

ippiki-ookami

303216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have proved that C is false after proving that (1) is true by contradiction. In the proof of (1), I have used (not C).
  – Ilyes
  Jul 27 at 17:07
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  When C is false both your (1) and (2) are logically equivalent. That means they will take the same truth values, so if you proved that one is true, so should the other be, do you agree ?
  – ippiki-ookami
  Jul 27 at 17:20
  

  
  
  
  

add a comment | 

up vote
0
down vote

If $C equiv 0$, then $B lor C equiv B lor 0 equiv$ B, so that $(A implies B) equiv (A implies (B lor C ))$. So if $(A implies (B lor C ))$ takes value $1$, so does $(A implies B)$, and conversely for $0$.

share|cite|improve this answer

answered Jul 27 at 14:46

ippiki-ookami

303216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have proved that C is false after proving that (1) is true by contradiction. In the proof of (1), I have used (not C).
  – Ilyes
  Jul 27 at 17:07
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  When C is false both your (1) and (2) are logically equivalent. That means they will take the same truth values, so if you proved that one is true, so should the other be, do you agree ?
  – ippiki-ookami
  Jul 27 at 17:20
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

If $C equiv 0$, then $B lor C equiv B lor 0 equiv$ B, so that $(A implies B) equiv (A implies (B lor C ))$. So if $(A implies (B lor C ))$ takes value $1$, so does $(A implies B)$, and conversely for $0$.

share|cite|improve this answer

answered Jul 27 at 14:46

ippiki-ookami

303216

If $C equiv 0$, then $B lor C equiv B lor 0 equiv$ B, so that $(A implies B) equiv (A implies (B lor C ))$. So if $(A implies (B lor C ))$ takes value $1$, so does $(A implies B)$, and conversely for $0$.

share|cite|improve this answer

answered Jul 27 at 14:46

ippiki-ookami

303216

share|cite|improve this answer

share|cite|improve this answer

answered Jul 27 at 14:46

ippiki-ookami

303216

answered Jul 27 at 14:46

ippiki-ookami

303216

answered Jul 27 at 14:46

ippiki-ookami

303216

303216

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have proved that C is false after proving that (1) is true by contradiction. In the proof of (1), I have used (not C).
  – Ilyes
  Jul 27 at 17:07
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  When C is false both your (1) and (2) are logically equivalent. That means they will take the same truth values, so if you proved that one is true, so should the other be, do you agree ?
  – ippiki-ookami
  Jul 27 at 17:20
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have proved that C is false after proving that (1) is true by contradiction. In the proof of (1), I have used (not C).
  – Ilyes
  Jul 27 at 17:07
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  When C is false both your (1) and (2) are logically equivalent. That means they will take the same truth values, so if you proved that one is true, so should the other be, do you agree ?
  – ippiki-ookami
  Jul 27 at 17:20
  

  
  
  
  

I have proved that C is false after proving that (1) is true by contradiction. In the proof of (1), I have used (not C).
– Ilyes
Jul 27 at 17:07

I have proved that C is false after proving that (1) is true by contradiction. In the proof of (1), I have used (not C).
– Ilyes
Jul 27 at 17:07

When C is false both your (1) and (2) are logically equivalent. That means they will take the same truth values, so if you proved that one is true, so should the other be, do you agree ?
– ippiki-ookami
Jul 27 at 17:20

When C is false both your (1) and (2) are logically equivalent. That means they will take the same truth values, so if you proved that one is true, so should the other be, do you agree ?
– ippiki-ookami
Jul 27 at 17:20

add a comment | 

up vote
0
down vote

Are you asking whether $Ato B$ is a valid inference from $Ato(Bvee C)$ and $neg C$?   It is.

Take $neg C$ and $Ato(Bvee C)$ as premises.   By assuming $A$, you may derive that $B$ or $C$ will be true, but $C$ is promised to be false, leaving only that $B$ may be true.   Therefore $Ato B$ is entailed by the premises.

So if you have proven $Ato (Bvee C)$ and have proven $neg C$, then you may infer that $Ato B$ is provable.

share|cite|improve this answer

answered Jul 27 at 22:18

Graham Kemp

80k43275

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I agree with the two responses. When we can't prove directly (1), The proof of (2) is an indirect proof of (1), if C is false.
  – Ilyes
  Jul 27 at 22:40
  

  
  
  
  

add a comment | 

up vote
0
down vote

Are you asking whether $Ato B$ is a valid inference from $Ato(Bvee C)$ and $neg C$?   It is.

Take $neg C$ and $Ato(Bvee C)$ as premises.   By assuming $A$, you may derive that $B$ or $C$ will be true, but $C$ is promised to be false, leaving only that $B$ may be true.   Therefore $Ato B$ is entailed by the premises.

So if you have proven $Ato (Bvee C)$ and have proven $neg C$, then you may infer that $Ato B$ is provable.

share|cite|improve this answer

answered Jul 27 at 22:18

Graham Kemp

80k43275

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I agree with the two responses. When we can't prove directly (1), The proof of (2) is an indirect proof of (1), if C is false.
  – Ilyes
  Jul 27 at 22:40
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

Are you asking whether $Ato B$ is a valid inference from $Ato(Bvee C)$ and $neg C$?   It is.

Take $neg C$ and $Ato(Bvee C)$ as premises.   By assuming $A$, you may derive that $B$ or $C$ will be true, but $C$ is promised to be false, leaving only that $B$ may be true.   Therefore $Ato B$ is entailed by the premises.

So if you have proven $Ato (Bvee C)$ and have proven $neg C$, then you may infer that $Ato B$ is provable.

share|cite|improve this answer

answered Jul 27 at 22:18

Graham Kemp

80k43275

Are you asking whether $Ato B$ is a valid inference from $Ato(Bvee C)$ and $neg C$?   It is.

Take $neg C$ and $Ato(Bvee C)$ as premises.   By assuming $A$, you may derive that $B$ or $C$ will be true, but $C$ is promised to be false, leaving only that $B$ may be true.   Therefore $Ato B$ is entailed by the premises.

So if you have proven $Ato (Bvee C)$ and have proven $neg C$, then you may infer that $Ato B$ is provable.

share|cite|improve this answer

answered Jul 27 at 22:18

Graham Kemp

80k43275

share|cite|improve this answer

share|cite|improve this answer

answered Jul 27 at 22:18

Graham Kemp

80k43275

answered Jul 27 at 22:18

Graham Kemp

80k43275

answered Jul 27 at 22:18

Graham Kemp

80k43275

80k43275

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I agree with the two responses. When we can't prove directly (1), The proof of (2) is an indirect proof of (1), if C is false.
  – Ilyes
  Jul 27 at 22:40
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I agree with the two responses. When we can't prove directly (1), The proof of (2) is an indirect proof of (1), if C is false.
  – Ilyes
  Jul 27 at 22:40
  

  
  
  
  

I agree with the two responses. When we can't prove directly (1), The proof of (2) is an indirect proof of (1), if C is false.
– Ilyes
Jul 27 at 22:40

I agree with the two responses. When we can't prove directly (1), The proof of (2) is an indirect proof of (1), if C is false.
– Ilyes
Jul 27 at 22:40

add a comment | 

 

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