Let $P(x)$ be an open sentence. Is "$P(x)$ and not $P(x)$ proposition? Let P(x) be an open sentence. Is "P(x) and not P(x)" a proposition ? And another question. Is  " if n=2, then n is even" a proposition ? P.S. I don't know where link of teaching for writing symbol of maths is. Help me please. Thank you. 
 A: Well, we seem to be dealing with first-order logic here, so I suppose you mean whether $P(x) \land \neg P(x)$ is a sentence.
In usual terminology, we define a sentence to be a well-formed first-order formula with no free occurrence of variables, i.e. a closed formula. Thus, $P(x) \land \neg P(x)$ is not a sentence in this sense.
Yes, "if $n=2$, then $n$ is even" can be regarded as proposition, particularly, a conditional one:

$P \rightarrow Q$

where $P$ represents '$n=2$' and $Q$ stands for '$n$ is even'.
A: "P(x) and not P(x)" is an open sentence as well. Why shouldn't it be? It has the variable x."
  No because it is FALSE for all possible values of x.  This statement is a proposition.
"The same holds for "if n=2, then n is even"." 
n is NOT a variable here because it is equal to 2.  This statement is true and so is a proposition.
Demosthenes, a "proposition" is a statement that is either true or false (even if you do not know which)O.  An "open sentence" is a statement containing a variable which may be true or false depending on the value of the variable.
