We studied propositional logic. Lets take a statement “x > 5” is this statement a proposition? The answer is no. Whenever the statements have variable(s) in them we cannot say those statements as a proposition. The question here is can we make such statements to propositions? The answer here is yes. In the above statement there are two parts one is the variable part called “subject” and another is relation part “>5” called “predicate”. We can denote the statement “x>5” by P(x) where P is predicate “>5” and x is the variable.
We also call P as a propositional function where P(x) gives value of P at x. Once value is assigned to the propositional function then we can tell whether it is true or false i.e. a proposition. For e.g. if we put the value of x as 3 and 7 then we can conclude that P(3) is false since 3 is not greater than 5 and p(7) is true since 7 is greater than 5. We can also denote a statements with more than one variable using predicate like for the statement “x = y” we can write P(x,y) such that P is the relation “equals to” . Similarly the statements with higher number of variables can be expressed. The logic involving predicates is called Predicate Logic or Predicate calculus.
Example:
- Let P(x): x+2<10, find the truth value of P(5) and P(9).
Solution: Given, P(x): x+2<10
When x=5, P(5) : 5+2<10
7<10 (True)
When x=9, P(9) : 9+2<10
11<10 (False)
Therefore, P(5) is true and P(9) is false for P(x): x+2<10.
2. Let F(x,y): x=y+6, find the truth value of F(1,5) and F(6,0).
Solution: Given, F(x,y): x=y+6
When x=1 and y=5, F(1,5): 1=5+6
1=11 (False)
When x=6 and y=0, F(6,0): 6=0+6
6=6 (True)
Therefore, F(1,5) is false and F(6,0) is true for F(x,y): x=y+6.