# Material Conditional Not Logically Equivalent?

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.

• Same error as in MatSE : ¬(a and ¬ b) is TRUE when a is FALSE and b is TRUE. Jul 15, 2014 at 15:34