I got screwed when trying to understand this expression. I've thought several times but I cant get the meaning.
! (p || q) is equivalent to !p && !q
For this one, somehow I can comprehend a little bit.
My understanding is " Not (p q) = not p and not q" which is understandable
! (p && q) is equivalent to !p || !q
For the second, I'm totally got screwed. How come
My understanding is " Not (p q) = Not p or Not q " . How come and and or can be equivalent each other? as for the rule in the truth table between && and || is different.
That's how I comprehend each expression, perhaps I have the wrong method in understand the expression. Could you tell me how to understand those expressions?
You can use a Truth table to see how the two expressions are equal. Like This:
!(P || Q) = !P && !Q
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P Q P || Q !(P||Q) !P !Q !P && !Q
_________________________________________________
1 1 1 0 0 0 0
1 0 1 0 0 1 0
0 1 1 0 1 0 0
0 0 0 1 1 1 1
_________________________________________________
Note that the column labeled !(P||Q) is the same as the column labeled !P && !Q. You can work this from the left most column where we set the initial values for P and Q. Then work out each column towards the right.
!(P && Q) = !P || !Q
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P Q P && Q !(P&&Q) !P !Q !P && !Q
_________________________________________________
1 1 1 0 0 0 0
1 0 0 1 0 1 1
0 1 0 1 1 0 1
0 0 0 1 1 1 1
_________________________________________________
Think of it in terms of the Red Toyota.
Let p = "The car is red"
Let q = "The car is a Toyota"
! ( p && q ) means "The car is not a red Toyota"
Which is the same as saying:
!p || !q "it's not red, or (inclusive) it's not a Toyota" , right?
