[xecure]

I received a question as homework which had asked me to explain why the negation of "Al and Bill are absent" is not "Al and bill are present". So I went about making a truth table for it and this is what I came up with:

A: Al is absent
B: Bill is absent

 A | B | A^B | ~A | ~B | ~A^~B
---+---+-----+----+----+-------
 T | T |  T  |  F |  F |  F
 T | F |  F  |  F |  T |  F
 F | T |  F  |  T |  F |  F
 F | F |  F  |  T |  T |  T

Looking at my last column the negation of "Al and Bill are present" is only true when both "Al is absent" is false and "Bill is absent" is false. I assumed if them both being absent is false, that they were both present. Following that thought, I don't understand how the negation of "Al and Bill are absent" cannot be "Al and Bill are present"

Can someone give me some insight?

[KYA]

Is it semantics? Al and Bill are present is equivalent to Al and Bill are not absent, English language wise; however, since we are talking in logical equivalences:


A = Al is absent
B = Bill is absent
~A = Al is present
~B = Bill is present

A AND B = Al and Bill are absent
~( A AND B ) according to DeMorgan's Laws becomes

~A OR ~B = Al or Bill is not absent

[mostyfriedman]

If you negate "Al and Bill are absent" as "Al and Bill are present", its too strong of a statement to state that they are both present, when in fact only one of them may have been present and the other absent. We can construct the truth table for the proposition as follows to clarify things

A = "Al is absent"
B = "Bill is absent"

the proposition "Al and Bill are absent" will have the following truth table
A B P=(A and
F F F
F T F
T F F
T T T

now if you negate the proposition "Al and Bill are absent", the truth values for P should get inverted in the previous table(every T should be F and vice versa). If we assume the correct negation for P is "Al and Bill are present" then we have the following truth table

A B (!A and !B )
F F T
F T F
T F F
T T F

See??, the column for the negation of P came out incorrect it should have been as follows

!P = (!A or !B )
T
T
T
F

So as you can see the correct negation of P is "Al or Bill is present" (remember its an inclusive or here).

So in short (!A and !B ) is inequivalent to !(A and B ), to find an equivalent expression for !(A and B ) you must use De Morgan's law as KYA showed you. hope that helped

This post has been edited by mostyfriedman: 12 October 2010 - 11:13 PM

[mostyfriedman]

also to convince yourself that the negated expression should be (!A or !B ) you can use K-map method to find a sum of products expressions for !P by taking the T rows.