DHARM138 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE
This expression is also called a predicate expression or predicate formula.
In true sense symbol “∀” quantifies the variable x therefore, it is called quantifier.
Let’s take another statement
“Some boys are good”
To translate the statement we required to symbolize the expression like “there exists
some x” or “few x” or “for some x”. For that we use the symbol “(∃x)” and this symbol is called
existential quantifier. Thus, the statement symbolize equivalently by the expression
(∃x) G(x) : read as “there exists some x such that x is good boy”
It must also be noted that, quantifier symbols (“∀” or “∃”) always be placed before the
statement function to which it states.
To make things more clear we illustrated few examples to symbolize the statements
using quantifiers.
Example 5.29
- “There are white Tigers”
We use the statement functions
T(x) : i.e., “x is Tiger”
and W(x) : i.e., “x is white”
Then, (T(x) ∧ W(x)) : translated as “x is white Tiger”. To translate the statement “There
are white Tigers” which is equivalent to the statement “There exists some white Tigers” or
“There are few white Tigers” we can write,
(∃x) (T(x) ∧ W(x))
Reader should not worry about the unique predicate expression for a statement. Possi-
bly, a statement can be translated into several different predicate expressions. Like if G(x) s.t.
“x is white Tiger” then predicate expression (∃x) G(x) is also a correct translation of the above
statement. - All human are mortal”
Assume, M(x) : i.e.,“x is mortal”
H(x) : i.e.,“x is human”
So, the sense of the statement “if human then mortal” can be translated using symbol
(H(x) → M(x)).
To symbolize “for all x”, quantify the variable x by introducing “(∀x)” and put before the
statement expression s.t.
(∀x) (H(x) → M(x))
(Expression is read as “for all x, if x is human then x is mortal”? “All human are mortal”) - “John is human”
Simply translated by H(j), where H be the predicate “human” and j is the name “John”. - “For every number there is a number greater than it”
The statement can be equivalently expressed by,
“For all x, if x is a number then there must exist another number (say y) such that y is
greater than x”.
Assume, G(x, y) : i.e., “y is greater than x”
N(x) : ,, “x is a number”