called a disproof of p In the next section you will read about some ways of disproving a statement Sometimes it happens that we feel a certain statement is true but we don t succeed in proving it It may also happen that we can t disprove it Such statements are called conjectures If and when a conjecture is proved it would be called a theorem If it is disproved then its negative will be a theorem In this context there s a very famous conjecture which was made by a mathematician Goldbach in 1742 He stated that For every n E N if n is even arid n 2 then n is the sum of two primes To this day no one has been able to prove it or disprove it To disprove it several people have been hunting for an example for which the statement is not truc i e an even number 1 2 such that h cannot be written as the sum of two prime numbers Now as you have seen a mathematical proof of a statement consists of one or more premises These premises could be of four types i a proposition that has been proved earlier e g to prove that the complex roots of a polynomial in R x occur in pairs we use tlie division algorithm or ii a proposition that follows logically from the earlier the proof as you have seen in Example 1 or iii a mathematical fact that has never been proved but is universally accepted as true e g two points determine a line Such a fact is called an axiom or a ppstulate Dositions given in iv the definition of a mathematical term e g assuming the definition of C in the proof of AnBCA You will come across more examples of each type while doing the following exercises and while going through proofs in this course and other courses El Write down an example of a theorem and its proof of at least 4 steps taken from school lcvcl algebra At each step indicate which of tlie four types of premise it is E2 Is every statement a theorem Why So far we have spoken about valid or acceptable arguments Now let us see an example of a sequence of statements that will not Sorm a valid argument Consider the following sequence If Maya sees the movie she won t finish her homework Maya won t finish her homework Therefore Maya sees the movie Looking at the argument can you say whether it is valid or not Intuitively you may feel that the argument isn t valid But is there a formal logical tool that you can apply to checlr if your intuition is correct What about truth tables Let s see The given argument is of the form pa q P where p Maya sees the movie and q Maya won t finish her homework J M