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Question 498974: What is the truth table for ( p˅q) → (p^q)?
What is the truth table for (p→q)↔ ~ r ?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! here's your truth tables.
the first truth table is for (pvq) -> (p^q)
you have 2 variables so you need 4 rows (2 * 2).
all possible conditions are shown in the truth table for the 2 variables.
(pvq) is true except when both p and q are false (FF).
all other conditions make it true (TT, TF, FT).
(p^q) is false except when both p and q are true (TT).
all other conditions make it false (TF, FT, FF).
(pvq) -> (p^q) is true except when (pvq) is true and (p^q) is false (TF).
all other conditions make it true (TT, FT, FF).
the second truth table is for (p->q) <-> ~r
you have 3 variables so you need 8 rows (2 * 2 * 2)
all possible combinations are shown in the truth table for the 3 variables.
(pvq) -> (p^q) is true except when (pvq) is true and (p^q) is false (TF).
all other conditions make it true (TT, FT, FF).
~r is true when r is false and ~r is false when r is true.
(p->q) <-> ~r is true when the truth tables for (p->q) and ~r are the same TT, FF).
(p->q) <-> ~r is falser when the truth tables for (p->q) and ~r are not the same (TF, FT).
in general, this is how the truth tables work.
A or B is true in all cases except when A and B are false.
A and B is false in all cases except when A and B are true.
A implies B is true in all cases except when A is true and B is false.
A if and only if B is true when A and B agree (either both true or both false) and is false when A and B disagree (one is true and the other is false).
not A is true if A is false and is false if A is true.
here's a truth table that shows all the possible combinations.
A B ~A A or B A and B A implies B A if and only if B
T T F T T T T
T F F T F F F
F T T T F T F
F F T F F T T
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