Mathematicalproving

Mathematicalproving

1. Direct method

   • If p, then q ( p ---> q )
   • Outline
     • If r is a, then s is b
       • 1) suppose r is a
         • 2) then r = definition of a
           • 3) s = definition of a
             • 4) expand
               • 5) factorise
                 • 6) relate s to definition of b
                   • 7) therefore s is b
   • Example :-
     • Is m is an even integer, then 7m+4 is an even integer
       • 1) suppose m is an even integer
         • 2) then m = 2n
           • 3) 7m+4 = 7(2n)+4
             • 4) = 14n+4
               • 5) = 2(7n+2)
                 • 6) = 2q, q=7n+2
                   • 7) therefore 7m+4 is an even integer

2. Indirect method

   • Contrapositive
   • Contradiction
     • Not limited to conditional statement
     • conditional statement
       • example :
         • if [n^2]
       • outline :
         • proposition : p ---> q
           • assume p is true, q is false ( p ---> ~q )
             • come across a contradiction
               • then p --> q must be true
     • non conditional statement
       • proposition : p
         • assume ~ p is true
           • come across a contradiction
             • then p must be true
   • Mathematical induction