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Section 1.7: Problem 5 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Where is a set of wffs, define a deduction from to be a finite sequence of wffs such that for each , either (a) is a tautology, (b) , or (c) for some and less than , is . (In case (c), one says that is obtained by modus ponens from and .) Give a deduction from the set the last component of which is .
Note that a deduction is a sequence of “reasoning” that allows to deduce the last term from the set of initial assumptions . In our case, is tautologically equivalent to , i.e. we need to deduce from , and . Here is one possible sequence: , , , , , , .