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There are many kinds of statement that are not theorems: A statement can be an axiom, that is, something that is assumed to be true without proof. It is usually self-evident, but like Euclid's parallel postulate, need not be. A statement need not be true in all circumstances - for example, A*B = B*A (commutativity) is not necessarily true for matrix multiplication. A statement can be false. A statement can be self-contradictory for example, "This statement is false".
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To prove a statement false, you need ONE example of when it is not true.To prove it true, you need to show it is ALWAYS true.
It is often a tautology - a statement which is true bit adds nothing to anyone's knowledge. A recent example is the fatuous remark by the British Prime Minister "Brexit is Brexit".
You make a statement that is either a true one or a false one. Simple as that. For example, you might say "The moon is made of green cheese" as a false statement and most people know that's wrong.