In Rosen's Discrete Maths text the example propositions seem the same as proposition functions? Very noob to math and stackexchange:
Example propositions given in Rosen's Discrete Maths textbook, such as
a. "It is raining today"; or
b. "Today is Thursday" -
Can't they be considered as propositional functions themselves:
If $x$ is a variable with all days in it, the truth value of "It is raining today" varies with $x$, and so is a propositional function of $x$ rather than a proposition, no?
Is my understanding correct and can I use it as a basis for further study?
 A: The CORE Distinction is whether the Statements have some variables or not.
Propositions : No variables. Maybe true or not true. The "truth value" will not change or will not Depend on variables.
Propositional functions : must have atleast 1 variable. May have 2 or more variables. In general, Will be neither "true" nor "not true" , until all the variables are assigned some values. In general, the "truth value" will change when the variables are assigned some other values.
Statements like "Noida is in India" & "India contains Noida" are Propositions, either true or not true.
Statements like "X is in Y" or "X contains Y" are Propositional functions with 2 variables, which become true or not true when we use some values of X & Y.
Eliminating only X (or only Y) will still give Propositional functions with 1 variable.
Eg "X is in India" & "India contains Y".
Eliminating both variables will give Propositions which are either true or not true.
When written in "English" , it may not be clear whether we are talking about all Items or variable Items.
Consider "$x+1=2$" where "$x$" is a variable : When we do not know "$x$" variable value, It is neither true nor not true. When we do know "$x$" variable value, It is either true (when $x=1$) or not true (when $x=3$) otherwise.
In case we know "$x=1$" a constant , then "$x+1=2$" is true & it is a Proposition because we are not making "$x$" a variable. It is almost like "$1+1=2$" (true) or like "$3+1=2$" (not true) where we have no variables.
Consider "Students who have taken calculus can attend this class" , which we can write like this : "for all S in Students : Student S has taken calculus IMPLIES Student S can attend this class"
$\forall S \in Students : Student\ S\ has\ taken\ calculus \implies Student\ S\ can\ attend\ this\ class$
Here, there are no variables to which we can assign values.
If it was "Student Z who has taken calculus can attend this class" , we can write it like this : "P(Z) : Z has taken calculus" & "Q(Z) : Z can attend this class" :
$P(Z) : Z\ has\ taken\ calculus$
$Q(Z) : Z\ can\ attend\ this\ class$
where $Z$ is a variable.
In that case, we can not know whether $P$ & $Q$ are true or not true until we assign a value to $Z$.
We are given that "P(Z) IMPLIES Q(Z)" :
$P(Z) \implies Q(Z)$
where $Z$ is a variable.
We are told that whatever value of $Z$ we take, it is always true, in this given context.
