2

Consider the set of predicates $M(x)$, $F(x)$, $S(x, y)$, and $P(x, y, z)$ with meanings “is male”, “is female”, “are siblings”, and “are parents of”, respectively.

Write a formula for predicate $C(x, y)$ which means $x$ and $y$ are cousins, that is one is the child of an aunt or uncle of the other.

How would I even go about that? Would I need to introduce a new variable? I am stumped.

fonini
  • 2,728
no_sleep
  • 77

3 Answers3

1

$x$ and $y$ are cousins when their parents are siblings.

This means that there exist $p_1$ and $p_2$ such that $p_1$ and $p_2$ are siblings, and $p_1$ is one of the parents of $x$ and $p_2$ is one of the parents of $y$.

Finally, the sentence "$p$ is one of the parents of $x$" means that there exists $q$ such that $p$ and $q$ are parents of $x$.

fonini
  • 2,728
  • How would your second part translate to predicates? Also, there are only three variables that I can use and you are using more than 3. – no_sleep Feb 02 '15 at 00:17
  • It translates just as it's written: $\exists q : P (p, q, x)$. If you want to limit the number of variables in the existence sentences, I think you can reuse the $ q $ for $ p_1$ and $ p_2$. – fonini Feb 02 '15 at 00:36
1

Hint: $\forall x,y,p_1,p_2,p_3,p_4:[P(p_1,p_2,x)\land P(p_3,p_4,y)\implies [C(x,y) \iff ???]]$

where $p_1$ and $p_2$ are the parents of $x$, and $p_3$ and $p_4$ are the parents of $y$.

$x$ and $y$ would be cousins, for example, if $p_1$ and $p_3$ are siblings, i.e. if $S(p_1,p_3)$. You will need to consider all possible combinations. Remember that $S$ is a symmetric relation.

Dan Christensen
  • 14,529
0

Hint. People $x$ and $y$ are cousins (the mention of aunts and uncles makes it clear that only first cousins are meant) if and only if one of $x$'s parents and one of $y$'s parents are siblings.

David
  • 82,662