Hi all this is my first time using this site so I hope I am presenting this properly, apologies if not. It is said that “a → b” and “¬(a and ¬ b)” are logically equivalent but I do not understand why/agree. However I agree that “a → b” and “¬a or b” are logically equivalent due to their truth tables being the same.
Here is why I do not understand (please point out where I made a mistake – I know I must have): The truth table for a → b is like this:
a b a → b
T T T
T F F
F T T
F F T
I can understand this and am happy with it. However this is now where I do not agree with what is accepted. The truth table for ¬(a and ¬b):
a b ¬(a and ¬ b)
T T T
T F F
F T (F)
F F T
Now I disagree on the third row and now I will show my working for that: ¬(a and ¬ b) in words is:
not(a and not b) insert truth values for a and b
not(F and not T) which simplifies to
not(F and F) which simplifies to
not(T) which simplifies to
F
This is why I think that it should be false on the third row which would disagree with the third row for a → b. I know I must have made a mistake/logical error but at the moment I cannot see it. I am new to studying logic so if anyone could point out what I have done wrong that would be great.
Let's recall the truth table definition for $\land$ (and):
$T \land T = T$
$T \land F = F$
$F \land T = F$
$F \land F = F$
$A \land \lnot B$ under the assignment $A = F, B= T$ is $$F \land \lnot T = \;F \land F = F$$
That makes $\lnot (A \land \lnot B)$ evaluate to $\lnot (F) = T$.