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Patrick from Woodbridge school sent us his
thoughts, which were artfully clear, in which he reduced each
sentence to a clear logical statement which could then be
NOTted:
Using de Morgan's law that NOT(A and B) = NOT(A) OR NOT(B) and
assuming that Bad means NOT(GOOD):
1. A good pet is friendly and furry is equivalent to
GOOD = Friendly AND Furry
so
NOT GOOD = NOT(Friendly AND Furry)
Using de Morgan's law
NOT GOOD = NOT(friendly) OR NOT(furry)
so d)
A bad pet (NOT good)
is
unfriendly (NOT
friendly)
OR unfurry (NOT
furry).
2. That man is lying or I'll eat my hat
Let X = Lying OR hat
NOT(X)
= NOT(lying OR hat)
= NOT[NOT(lying) AND NOT(hat)]
= NOT(truth AND NOT(hat))
so A. That man is telling the truth and I won't eat my hat
3. If you don't go to the party and if John goes to the party then
I won't go to the party
IF [NOT(you) AND john] THEN NOT(me)
so IF NOT[NOT(you) AND john] THEN NOT[NOT(me)] IF you OR NOT(john)
THEN me
So, the answer is:
If you go to the party, or if John does not go to the party, then I
will go to the party.
4. Twas brillig, and the slithy toves Did gyre and gimble in the
wabe.
Let X = brillig AND gyre AND gimble
Then
NOT(X)
= NOT(brillig AND gyre AND gimble)
= NOT(brillig) OR NOT(gyre) OR NOT(gimble)
so Twasn't brillig, or the slithy toves did not gyre or gimble in
the wabe.
Steve used truth tables to work out the
negations, making use of NOT(X) is True if and only if X
is false
Enumerating the possible combinations for friendly/furry we see
that a good pet corresponds to a single row in the truth table. A
bad pet is found by negating this.
Friendly |
Furry |
Good Pet |
Bad Pet |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
We can see that this corresponds to 'A bad pet is not friendly or
not furry' because it has the same truth table values as the Bad
Pet column:
Friendly |
Furry |
Not Friendly |
Not Furry |
(Not Friendly) OR (Not Furry) |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
For the second part, it is confusing because going/not going are
opposites. To be clear, the statement is
X: If you don't go to the party and if John goes to the party then
I won't go to the party.
This is a little ambiguous because it does not say that I will go
to the party in all other cases. I will assume that this is the
case, which quickly gives the truth table for me going as:
You go |
John goes |
You don't go |
I go (according to rule X) |
0 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
You go |
John Goes |
I go (according to rule NOT(X)) |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
There is only 1 case in which I go now. This is logically the same
as
If you go to the party, or if
John does not go to the party, then I will go to the party.
It is also the same as
I won't go
if John doesn't go or if you go.