Criteria for Use :
Criteria for Use Does the problem require a proof for numbers in an infinite countable set?
Natural numbers? Positive non-zero Integers?
Least Element?
Three Step Process :
Three Step Process Show that the equation works for the first element of the set in question. Usually the set of Natural Numbers.
Establish a hypothesis. Make a generalized assumption for the nth element of the set.
Prove the assumption works for the nth plus one term.
Prove n + n = 2n. :
Prove n + n = 2n. For n =1, we have 1+1=2(1)↔2=2.
Assume that n+n=2n for any n in the natural numbers.
For (n+1), we have (n+1) + (n + 1) = 2(n+1)
n + 1 + n + 1 = 2n + 2
n + n + 1 + 1 =
2n + 2 = 2n + 2. 2=2. We can continue.
Assumption follows from observation in i.
Distributive Property
Commutative Property
We have established n+n=2n. Q.E.D.
Elements of a Proof :
Elements of a Proof Start your proof by writing “Proof:”
Write out your proof or show us your arguments and logic.
Indicate you have finished your proof by writing QED.
QED = Quod Est Demonstratum