Question 475363: Construct a truth table for (p Λ ~ q) ↔ q
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! here's your truth table:
p ~q (p^~q) q (p^~q)<->q
T T T F F
T F F T F
F T F F T
F F F T F
Your AND relationship (p^~q) requires both p and ~q to be true in order for it to be true.
That's why you only see one T in the (p^~q) column (third column).
The IF AND ONLY IF relationship, otherwise known as the EQUIVALENCE relationship, requires both (p^~q) and q to be True or both (p^~q) and q to be False in order for it to be true. Any other combination results in False.
Since (p^~q) and q can never both be true at the same time, then this relationship will only be true if they're both false.
You can see that (p^~q)<->q is true when both (p^~q) and q are both false.
The rules of the IMPLIES statement (a->b) and the EQUIVALENCE statement (a<->b) statements are difficult to understand.
Why a->b is true when a is false regardless of whether b is true or false is a difficult concept to grasp.
Similarly, why a<->b is true if both a and b are false is also a difficult concept to grasp.
The recommendation is to follow the rules regardless of whether you fully understand the concept behind them.
That's what I did above.
Hopefully I did it right.
Here's a reference on conditional statements that might help you to understand better.
http://www.rwc.uc.edu/koehler/comath/21.html
scroll down to IMPLIES and EQUIVALENCE you will see their truth tables.
The IMPLIES truth table is true in all cases except the case where the first variable is true and the second variable is false.
In that case, the implies statement is false.
It is true if both variable are true.
It is also true if the first variable is false regardless of the value of the second variable.
In the IMPLIES statement, the first variable is called the precedent and the second variable is called the conclusion.
The EQUIVALENCE truth table is true only when the first variable and the second variable agree.
They can both be true or they can both be false.
If they do not agree, i.e. one is true and the other is false, then the EQUIVALENCE statement is false.
The EQUIVALENCE STATEMENT is equivalent to (no pun intended) a->b AND b->a
This would show up as:
(a->b)^(b->a)
You have to buy the logic in the implies statement in order to buy the logic in the equivalence statement.
Once you buy the logic in the implies statement, than the logic in the equivalence statement can be shown through the use of the truth tables.
Here's a truth table showing you the relationship between the two.
a b a->b b->a a->b^b->a a<->b
T T T T T T
T F F T F F
F T T F F F
F F T T T T
Remember that a->b is true if a is false regardless of the value of b.
Remember that b->a is true if b is false regardless of the value of a.
The only time a->b is false is if a is true and b is false.
The only time b->a is false is if b is true and a is false.
This truth table shows that a<->b is equivalent to a->b^b->a
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