## Problem set 6

Assigned: Mar. 25

Due: Apr. 1

### Problem 1

Consider the first order language, with a domain of people and instants
of time and the following non-logical symbols:
B(u,v) --- Time u is before time v.

D(p,t) --- Person p is dead at time t.

L(p,t) --- Person p is alive at time t.

R(p,q) --- Person p is a parent of person q.

U(p,t) --- Person p is not yet born at time t.

A -- Amy

N --- Now (a time).

Express the following sentences:
- A. All of Amy's parents are alive now.
- B. Amy will have a child, not yet born.
- C. Everyone was once unborn, and everyone is eventually dead. (You do
not have to express the time relations here; just state that there is some
time when they are unborn and some time when they are dead.)
- D. If person p is alive at time u and dead at time v, then u is before v.
- E. Once you are dead, you stay dead.
- F. If p is a parent of q, then there is some time when p is alive and q
is not yet born.

### Problem 2

Let S be the sample space with 8 elements: S = { a,b,c,d,e,f,g,h }.

Let W be the event {a,b,c}; let X be the event {a,d,f,g}; let
Y be the event {b,f,g} and let Z be the event {e,g}.

Let P(a)=0.2; P(b)=0.10; P(c)=0.10; P(d)=0.15;
P(e)=0.14; P(f)=0.09; P(g)=0.06; P(h)=0.16.
A. Compute the values of P(W), P(X), P(Y), P(Z), P(W,X), P(Y,Z),
P(X|W), P(W|X), P(Y|W), P(Z|W), P(Y|X), P(Z|X), P(Z|X,Y), P(Y,Z|X).

B. Are W and X absolutely independent? Are Y and Z absolutely independent?
Are Y and Z conditionally independent given X?