Predicate vs Proposition in Logical Mathematics

A predicate is a statement that has variables, such as x + 5 = 12. A proposition does not have variables in its statement, such as 7 + 5 = 12. The difference between a predicate and a proposition is that predicates have variables, and propositions don’t. 

In a predicate statement, we can change the value of x by passing it as an input to a function. When x is supplied as a function input, such as F(x), where F() is the function and x is the input, the statement is read as “F of x.”

Predicate vs proposition example
"x is an integer" this statement has x as a placeholder
F(x): "x is an integer" this is a predicate because it has variables in its statement
F(5): "5 is an integer" x = 5 and F(x) becomes a proposition

A function takes one or more input variables and returns a true or false value. When the input variables of a function are supplied, the statement changes from F(x): "x is an integer" to F(5): "5 is an integer" and from a predicate to a proposition.

But we can’t substitute any random value for x because sometimes it will not make sense. If x = g, the predicate F(x): "x is an integer" becomes F(g): "g is an integer," which is false but also opens the door for supplying an endless of meaningless input. We need to identify the domain of x so that all the supplied input is valid.

The last example shows the importance of knowing the accepted input values of x, known as the domain. If the problem at hand says that the domain of x is all natural numbers, we know that anything other than natural numbers is not accepted as input, and therefore not in the domain of x.

Other names for the domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range—Wikipedia.

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