Truth table (simple proof) (1 Viewer)

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1. The problem statement, all variables and given/known data

prove that:
not A implies B
if and only if
not B implies A

2. Relevant equations

construct truth table

3. The attempt at a solution

the answer is given as a table (T means true, F means false):

A| B| not A implies B| not B implies A| IFF

T T T T T
T F T T T
F T T T T
F F F F T


I understand that since the 3rd and 4th columns are the same, this completes the proof. BUT, i dont understand when to put T and when to put F.
any help wud be very much appreciated.
Thank you
 
31,919
3,891
1. The problem statement, all variables and given/known data

prove that:
not A implies B
if and only if
not B implies A

2. Relevant equations

construct truth table

3. The attempt at a solution

the answer is given as a table (T means true, F means false):

A| B| not A implies B| not B implies A| IFF

T T T T T
T F T T T
F T T T T
F F F F T


I understand that since the 3rd and 4th columns are the same, this completes the proof. BUT, i dont understand when to put T and when to put F.
any help wud be very much appreciated.
Thank you
I assume your question is about what to put in the 3rd and 4th columns.

The only combination of truth values for which the implication A ==> B is false, is when the hypothesis (A here) is true but the conclusion (B here) is false.

It's the same for the implication ~A ==> B. The only combination for which this implication is false is when the hypothesis (~A) is true, but the conclusion (B) is false.
For ~A to be true, it must be that A is false, so looking at the first two columns of your truth table, the row that makes ~A ==> B false is the fourth row, where A is false and B is false.

The explanation for ~B ==> A is similar.
 
764
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''The only combination of truth values for which the implication A ==> B is false, is when the hypothesis (A here) is true but the conclusion (B here) is false.
''

But, if A is true, then B is false, why would this make A==>B false?
wouldnt A==>B be false if A is true AND B is true ?

Thank you
 

HallsofIvy

Science Advisor
41,626
821
If you believe that "if true then true" is a false statement, then you need to go back and review basic definitions.
 
764
0
OH...right i see. sorry i misunderstood. i read it as:

''The only combination of truth values for which the implication (not )A ==> B is false, is when the hypothesis (A here) is true but the conclusion (B here) is false''
 

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