Introduction to Proof Structure

Welcome to the world of proofs! This week, we were introduced to the structure of how to compose a proof. We are given a statement and required to write a specific step by step structure that demonstrates how we are supposed to carry about writing the proof itself. Comments were also required to show that you have understanding of the claims that you are making in the proof.

Below, I will demonstrate with an example on how a proof structure is to be written for this course with a few additional comments.

             

(for all)x E R, (for all)y E R, x<y => (there exists)z E R, x<z and z<y

Assume x E R, y E R   # this is the domain assumption (can only be made for universal quantifiers)

     

   Assume x < y # antecedent

      

      Let z’ …… # since the value of z has not been determined yet, it is z prime.

                       # this assumption depends on the values of x and y.

     [proof of z’ E R]

   

     Then z’ E R # this is written once this statement is actually proven

   

     [proof of x<z’ and z’ < y]

    

     Then x < z’ and z’ < y

   

     Then (there exists) z E R, x < z and z< y # introduce (there exists symbol)

     Then x < y => ex z’ E R # intro =>(implication)

Then (for all)x E R, (for all)y E R, x < y => (there exists)z E R, x<z and z<y #intro (for all)