Solutions to Chapter 32 Semantic concepts

A. Given the following interpretation:

domain:

people

$W(x)$:

blank _x is a woman (or girl)

$M(x)$:

blank _x is a man (or boy)

$Y(x,y)$:

blank _x is younger than blank _y

$O(x,y)$:

blank _x is an offspring of blank _y

$d$:

Dana

$a$:

Alex

express the following relations:

1. 1.
   ​

   blank _x is older than blank _y $Y(y,x)$

2. 2.
   ​

   blank _x is blank _y’s mother $O(y,x)\wedge W(x)$

3. 3.
   ​

   blank _x and blank _y are siblings (note that you can’t be your own sibling) $\exists z(O(x,z)\wedge O(y,z))\wedge \neg x=y$

4. 4.
   ​

   blank _x is blank _y’s brother $\exists z(O(x,z)\wedge O(y,z))\wedge \neg x=y\wedge M(x)$

B. Using the symbolization key from the previous exercise, symbolize the following sentences:

1. 1.
   ​

   Alex and Dana are sisters. $\exists z(O(a,z)\wedge O(d,z))\wedge \neg a=d\wedge W(a)\wedge W(d)$

2. 2.
   ​

   Fathers are older than their children. $\forall x\forall y((O(x,y)\wedge M(y))\to Y(x,y))$

3. 3.
   ​

   Alex’s parents are the same age. $\forall x\forall y((O(x,a)\wedge O(y,a))\to \neg Y(x,y))$