Proofs Using vOut
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Proofs Using vOut 1. V v -V A 2. V>-B A 3. -V>-N A -Bv-N GOAL A v B A>G B>G G A v B G Set up the vOut Strategy
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Proofs Using vOut. A v B A > G B > G G. A v B. G. 1. V v -VA 2. V > -BA 3. -V > -NA -Bv-NGOAL. Set up the vOut Strategy. Proofs Using vOut. A v B A > G B > G G. A v B. G. 1. V v -V A 2. V > -BA 3. -V > -NA -Bv-N GOAL. - PowerPoint PPT Presentation
Transcript of Proofs Using vOut
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A
-Bv-N GOAL
A v BA>GB>GG
A v B
G
Set up the vOut Strategy
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A
-Bv-N GOAL
A v BA>GB>GG
A v B
G
... by making the missing conditionals new goals.
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A
V>(-Bv-N) GOAL
-V>(-Bv-N) GOAL -Bv-N 1,?,? vO
A v BA>GB>GG
A v B
G
Prove the top goal using the >I Strategy
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A4. V PA
-Bv-N GOAL V>(-Bv-N) 4-? >I
-V>(-Bv-N) GOAL -Bv-N 1,?,? vO
A v BA>GB>GG
A v B
G
Apply >O whenever you can.
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A4. V PA -B 2,4 >O -Bv-N GOAL V>(-Bv-N) 4-? >I
-V>(-Bv-N) GOAL -Bv-N 1,?,? vO
A v BA>GB>GG
A v B
G
... and your goal comes by vIn
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A4. V PA5. -B 2,4 >O6. -Bv-N 5 vI7. V>(-Bv-N) 4-6 >I
-V>(-Bv-N) GOAL -Bv-N 1,7,? vO
A v BA>GB>GG
A v B
G
Now use >In to obtain your second goal.
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A4. V PA5. -B 2,4 >O6. -Bv-N 5 vI7. V>(-Bv-N) 4-6 >I8. -V PA
-Bv-N GOAL -V>(-Bv-N) 8-? >I -Bv-N 1,7,? vO
A v BA>GB>GG
A v B
G
Apply >O whenever you can.
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A4. V PA5. -B 2,4 >O6. -Bv-N 5 vI7. V>(-Bv-N) 4-6 >I8. -V PA -N 3,8 >O -Bv-N GOAL -V>(-Bv-N) 8-? >I -Bv-N 1,7,? vO
A v BA>GB>GG
A v B
G
... and your goal comes by vIn
Proofs Using vOut1. V v -V A2. V>-B A3. -V>-N A4. V PA5. -B 2,4 >O6. -Bv-N 5 vI7. V>(-Bv-N) 4-6 >I8. -V PA9. -N 3,8 >O10. -Bv-N 9 vI11. -V>(-Bv-N) 8-10>I12. -Bv-N 1,7,11 vO
A v BA>GB>GG
A v B
G
The proof is now complete.
Proofs Using vOut1. A&(BvC) A
(A&B)v(A&C) GOAL
A v BA>GB>GG
A v B
G
Do &Out whenever you can
Proofs Using vOut1. A&(BvC) A2. A 1 &O3. B v C 1 &O
(A&B)v(A&C) GOAL
A v BA>GB>GG
A v B
G
Set up the vOut Strategy
Proofs Using vOut1. A&(BvC) A2. A 1 &O3. B v C 1 &O
(A&B)v(A&C) 3,?,? vO
A v BA>GB>GG
A v B
G
Set the missing conditionals as new goals.
Proofs Using vOut1. A&(BvC) A2. A 1 &O3. B v C 1 &O
B >[(A&B)v(A&C)] GOAL
C >[(A&B)v(A&C)] GOAL (A&B)v(A&C) 3,?,? vO
A v BA>GB>GG
A v B
G
Use >In to prove the first goal.
Proofs Using vOut1. A&(BvC) A2. A 1 &O3. B v C 1 &O4. B PA
(A&B)v(A&C) GOAL B >[(A&B)v(A&C)] 4-? >I
C >[(A&B)v(A&C)] GOAL (A&B)v(A&C) 3,7,? vO
A v BA>GB>GG
A v B
G
Prove A&B so that you can use vIn to obtain the goal.
Proofs Using vOut1. A&(BvC) A2. A 1 &O3. B v C 1 &O4. B PA5. A&B 2,4 &I (A&B)v(A&C) GOAL B >[(A&B)v(A&C)] 4-? >I
C >[(A&B)v(A&C)] GOAL (A&B)v(A&C)
A v BA>GB>GG
A v B
G
Now the subproof can be completed with vIn.
Proofs Using vOut1. A&(BvC) A2. A 1 &O3. B v C 1 &O4. B PA5. A&B 2,4 &I6. (A&B)v(A&C) 5 vI7. B >[(A&B)v(A&C)] 4-6 >I
C >[(A&B)v(A&C)] GOAL (A&B)v(A&C) 3,7,? vO
A v BA>GB>GG
A v B
G
Now set up the >In Strategy to prove the second goal.
Proofs Using vOut1. A&(BvC) A2. A 1 &O3. B v C 1 &O4. B PA5. A&B 2,4 &I6. (A&B)v(A&C) 5 vI7. B >[(A&B)v(A&C)] 4-6 >I8. C PA
(A&B)v(A&C) GOAL C >[(A&B)v(A&C)] 8-? >I (A&B)v(A&C) 3,7,? vO
A v BA>GB>GG
A v B
G
This subproof can be completed in the same way.
Proofs Using vOut1. A&(BvC) A2. A 1 &O3. B v C 1 &O4. B PA5. A&B 2,4 &I6. (A&B)v(A&C) 5 vI7. B >[(A&B)v(A&C)] 4-6 >I8. C PA9. A&C 2,8 &I10. (A&B)v(A&C) 9 vI11. C >[(A&B)v(A&C)] 8-10 >I12. (A&B)v(A&C) 3,7,11 vO
A v BA>GB>GG
A v B
G
The proof is now complete.
