Predicate logic: Let the predicate p over the set of people M be defined as: p(x,y) := x is a parent of y.

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Marcel1104

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Predicate logic: Let the predicate p over the set of people M be defined as: p(x,y) := x is a parent of y. What is sought is the predicate logical expression for
1. Not every person has a child.
2. Every person has a gradparent.
3. There are people who don’t have siblings.

I thought about:
1.
{\displaystyle \lnot }
(
{\displaystyle \exists }
X
{\displaystyle \in }
M:
{\displaystyle \forall }
y
{\displaystyle \in }
M: p(x,y))
2.
{\displaystyle \forall }
x
{\displaystyle \in }
M:
{\displaystyle \exists }
y
{\displaystyle \in }
M: p(p(y,x),x)
3.
{\displaystyle \exists }
X, y
{\displaystyle \in }
M:
{\displaystyle \forall }
z
{\displaystyle \in }
M:
{\displaystyle \lnot }
(p(z,x) v p(z,y))

Is it correct?:)
 

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Marcel1104 said:
Predicate logic: Let the predicate p over the set of people M be defined as: p(x,y) := x is a parent of y. What is sought is the predicate logical expression for
1. Not every person has a child.
2. Every person has a gradparent.
3. There are people who don’t have siblings.

I thought about:
1.
{\displaystyle \lnot }
(
{\displaystyle \exists }
X
{\displaystyle \in }
M:
{\displaystyle \forall }
y
{\displaystyle \in }
M: p(x,y))
Click to expand...

This seems to be the negation of "There exists a person who is the parent of everybody in the set (including him/herself?). Wouldn't the negation be "For all people, there is somebody who is not that person's child"? (I'm a little rusty on this kind of logic, so if somebody else replies with something else, go with that.)

Marcel1104 said:
2.
{\displaystyle \forall }
x
{\displaystyle \in }
M:
{\displaystyle \exists }
y
{\displaystyle \in }
M: p(p(y,x),x)
Click to expand...

By definition of p(x,y), isn't this saying that p(y,x) is the parent of x? Should maybe there be a third variable to indicate the parent (that is, the person between the grandparent and the child)?

Marcel1104 said:
3.
{\displaystyle \exists }
X, y
{\displaystyle \in }
M:
{\displaystyle \forall }
z
{\displaystyle \in }
M:
{\displaystyle \lnot }
(p(z,x) v p(z,y))
Click to expand...

I think this will work.
 
Marcel1104 said:
3.
{\displaystyle \exists }
X, y
{\displaystyle \in }
M:
{\displaystyle \forall }
z
{\displaystyle \in }
M:
{\displaystyle \lnot }
(p(z,x) v p(z,y))
Click to expand...
To me this says that there exist two people, i.e. 'x' and 'y' neither of whom has a parent. I wonder if this is a typo since you only need to change one operation in there to make it correct.
 
blamocur said:
To me this says that there exist two people, i.e. 'x' and 'y' neither of whom has a parent. I wonder if this is a typo since you only need to change one operation in there to make it correct.
Click to expand...
Actually I am wrong too: my change would say that one can find a pair of people who aren't siblings. This shows that I am pretty rusty on predicate logic too :(
 
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