DHARM136 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE5.7.1 Symbolization of Statements using Predicate..................................................
Since, predicate is used to describe the feature of the statement, therefore a statement could
be written symbolically in terms of the predicate symbol followed by the name to which, predicate
is applied.
To symbolize the predicate and the name of the object we shall use following convention,
· A predicate is symbolize by a capital letter (A, B, C, .......Z).
· The name of the individual or object by small letter (a, b, c, .......z).
For example, consider the statement
“Rhodes is a good boy”
where the predicate is “good boy” and denoted symbolically by the predicate symbol G
and the name of the individual “Rhodes” by r. Then the statement can be written as G(r)
and read as “r is G”.
Similarly the statement “Stephen is a good boy” can be translated as G(s) where s stands
for the name “Stephen” and the predicate symbol G is used for “good boy”.
To translate the statement “Stephen is not a good boy” that is the negation of the previ-
ous statement which can be written as ~ G(s). In the similar sense it is possible that name of
the individual or objects may varies for the same predicate. In general, any statement of the
type “r is S” can be denoted as S(r) where r is the object and S is the predicate.
As we said earlier that every predicate describes something about one/more objects. Let
we define a set D called domain set of universe (never be empty). From the set D we may take
a set of objects of interests that might be infinite. Let’s consider the statement,
G(r) : where r is a good boy
then, G ⊆ D
That can be described as, G = {r ∈ D / r is a good boy}
Since such type of predicate requiring single object is called one- place predicate.
When the number of names of the object associated with a predicate are two to form a
statement then predicate is two-place predicate. In true sense, the statement expressed by two
place predicate there exist a binary relation between the associated names. For example the
statement,
G(x, y) : (where x is greater than y) is a two-place predicate
then, G ⊆ D × D; where G consists of sets of pairs
where, set G = {(x, y) ∈ D / x > y}. For example if D is the set of positive integers (I+) then
G = {(2, 1), (3, 1), (3, 2), ........}.
Similarly, we can define a three-place predicate, for example the statement
P(x, y, z) : (where y and z are the parents of x)
then, P ⊆ D × D × D s.t. (a, b, c) ∈ P
In general, a predicate with n objects is called n-place predicate.
then, P ⊆ D × D × D............× D, n times
s.t. (t 1 , t 2 , t 3 , .....................tn) ∈ P
The truth values of the statement can also be determined on the basis of domain set D.
Assume set D is defined as,
D = {1, 2, 3, 4}