Section 1.6
Proofs Involving Quaitifiers
To prove a propositio n of the form (x)P(x) we must show P(x)
is true for every x in the universe
In the previous proof we have two assumption:
 Let x be an integer (Let x  Z )

Let x be even
CONSTRUCTIVE PROOF OF
( x ) p ( x )
In a constructive proof we only choose an element c in
the universe for which p(c) is true.
Valid Statements
Invalid Statements:
A proof by Counterexample
Exercises for sec 1.7
1 a,e,
2 a,b,c,
4 a,b,c,d
8a