likely to turn around look menacingly at me and say Prove it What she or he really wants is to be convinced of my statement by some evidence In this case it would probably be a big physical push Convincing evidence is also what the world asks for before accepting a scientist s predictions or a historian s claims In the same way if you want a mathematical statement to be accepted as true you would need to provide mathematically acceptable evidence to support it This means that you would need to show that the statement is universally true And this would be done in the form of a logically valid argument Definition An argument in mathematics or logic is a finite sequence of statements P P p such that p A P A A p p Each statement in the sequence except the last one i e pi for i 1 n is called a premise or an assumption or a hypothesis The final statement p is called the conclusion Let s consider an example of an argument that shows that a given statement is true Example 1 Give an argument to show that the mathematical statement For any two sets A and B An BCA is true Solution One argument could be the following Let x be an arbitrary element of AnB Then X E A and x E B by definition of n Therefore x A This is true for every x in An B Therefore AnBC A by definition of C The argument in Example 1 has a peculiar nature The truth of each of tho 4 premises and of its conclusion follows from the truth of the earlier premises in it Of course we start by assuming that the first statement is true Then assuming the definition of intersection the second statement is true Tlic third one is true whenever the second one is true because of the properties of logical implication The fourth statement is truc whenever the first three arc true because of the definition and properties of the term for all And finally the last statement is true whenever all the earlier ones are In this way we have shown that the given statcinent is true In other words we have proved the given statement as the following definitions show Definitions We say that a proposition p follows logically from propositions P P2 and Pn if p must be true whenever P P2 Pn are true i e P AP A AP p Here note the use of the implication arrow For any two propositions r and s rs denotes s is true whenever r is true Note that using the contrapositive this also denotes r is false whenever s is false Thus rs and rs are different except when both r and s are true or both are false A proof of a proposition p is a mathematical argument consisting of a sequence of statements P P2 Pn from which p logically follows So p is the conclusion of this argument The statement that is proved to be true is called a theorem Sometimes as you will see in Sec 2 3 3 instead of showing that a statement p is true we try to prove that it is false i e that p is true Such a proof is