# Tree of numbers

## Definition

In a sentence of the form [S [NP D CN] VP] the set A of entities denoted by the common noun CN can be divided into a subset with elements that belong to the set B of entities represented by VP, and a subset with elements that don't belong to that set, i.e. A intersect B and A - B, respectively. In a domain with n dogs, the dogs can be divided over these two subsets in n+1 ways, each of which is represented by an ordered pair x,y where x = |A intersect B| and y = |A - B|.

## Examples

The tree of numbers is a complete representation of all these pairs of numbers for each possible size of A:

```(i) |A|=0		       0,0
|A|=1		    1,0   0,1
|A|=2		 2,0   1,1   0,2
|A|=3	      3,0   2,1	  1,2	0,3
|A|=4          4,0   3,1   2,2   1,3   0,4
|A|=5       5,0   4,1   3,2   2,3	1,4   0,5
...	 		       ...
```

The meaning of a determiner D can be represented as a subset of a tree of numbers. The determiner every, for example corresponds to the x,0 pairs on each row:

```(ii) |A|=0		  +
|A|=1	      +	     -
|A|=2	   +  	  -	 -
|A|=3	+     -	      -	    -
...		 ...
```

Many properties of determiners (like upward monotonicity and downward monotonicity) and relations between determiners (like negation) can be clarified in the tree of numbers.