The output would be Logical Implications: A => B and B => A.

  • Thread starter John112
  • Start date
  • Tags
    implication
In summary, the implication A => B and B => A will both be true unless there is a combination of truth values for p and q where the implication is False. This means that A logically implies B and B logically implies A. A and B only do not imply each other if there is a case where one is True and the other is False. Therefore, the final column of the truth table for A => B and B => A does not necessarily have to resemble the truth table for an implication.
  • #1
John112
19
0
A: (p => ~q) ^ (p v q)
B: ~p v q

does A => B (A implies B) ?
does B => A ( B implies A) ?

I did the truth tables for each:

A => B:
http://www4c.wolframalpha.com/input/?i=((p+=>+NOT+q)+AND+(p+OR+q))+=>+(NOT+p+OR+q)

B => A:
http://www.wolframalpha.com/input/?i=(NOT+p+OR+q)+=>+((p+=>+NOT+q)+AND+(p+OR+q))+.

for A to logically imply B or for B to logically imply A, does it need to be a tautology? Meaning if A=> B or B=> A , on the truth tables should it be All Ts? or should the final column of the truh table of A=> B or B => A resemble the truth table of an implication?
 
Last edited:
Physics news on Phys.org
  • #2
John112 said:
A: (p => ~q) ^ (p v q)
B: ~p v q

does A => B (A implies B) ?
does B => A ( B implies A) ?

I did the truth tables for each:

A => B:
http://www4c.wolframalpha.com/input/?i=((p+=>+NOT+q)+AND+(p+OR+q))+=>+(NOT+p+OR+q)

B => A:
http://www.wolframalpha.com/input/?i=(NOT+p+OR+q)+=>+((p+=>+NOT+q)+AND+(p+OR+q))+.

for A to logically imply B or for B to logically imply A, does it need to be a tautology? Meaning if A=> B or B=> A , on the truth tables should it be All Ts? or should the final column of the truh table of A=> B or B => A resemble the truth table of an implication?

For the implication A => B, the only False you can have is when A is True and B is False. For all other combinations of truth values for A and B, the implication is considered to be True.
For the implication B => A, the only False you can have is when B is True and A is False. For all other combinations of truth values for B and A, the implication is considered to be True.
 
  • #3
Mark44 said:
For the implication A => B, the only False you can have is when A is True and B is False. For all other combinations of truth values for A and B, the implication is considered to be True.
For the implication B => A, the only False you can have is when B is True and A is False. For all other combinations of truth values for B and A, the implication is considered to be True.

So does A logically imply B? or does B logically implies A?
 
  • #4
Make a truth table with five columns, one each for p, q, (p => ~q) ^ (p v q), ~p v q, and ((p => ~q) ^ (p v q)) => ~p v q. You can say that A => B if the only false value you get in the fifth column is in the row where there's a T in the third column and an F in the fourth column.

Similar idea for B => A.
 

FAQ: The output would be Logical Implications: A => B and B => A.

1. How do you prove an implication?

To prove an implication, you must show that the antecedent (the "if" statement) implies the consequent (the "then" statement). This can be done through direct proof, contrapositive proof, or proof by contradiction.

2. What is the difference between direct and contrapositive proof?

In a direct proof, you start with the antecedent and use logical reasoning to arrive at the consequent. In a contrapositive proof, you assume the negation of the consequent and use logical reasoning to arrive at the negation of the antecedent. Both methods ultimately prove the same implication, but the approach may differ.

3. Can an implication be proven with a counterexample?

No, a counterexample can only disprove an implication. To prove an implication, you must provide a logical argument that shows the truth of the statement for all possible cases.

4. Is it possible to prove an implication with a truth table?

Yes, a truth table can be used to prove an implication. If all the possible combinations of truth values for the antecedent and consequent result in a true statement, then the implication is proven to be true.

5. How do you know when an implication is proven?

An implication is proven when there is a valid logical argument that shows the truth of the statement for all possible cases. This can be through direct, contrapositive, or proof by contradiction methods. Additionally, a truth table can also be used to prove an implication.

Back
Top