1. Aids to Formulating and Answering Key Questions NameWhen to use u Construction Methodproving “there exists” u Choose Method proving “for every” u Math Induction proving “for every n” u Existence Method using “there exists” u Specialization Methodusing “for every”

2. The Construction Method u Use when the goal is a “there exists” statement u Write the “there exists” in standard form: B: There exists an object with a certain property such that something happens u Construct the desired object - how ??? - often use Forward Process to construct u Prove the object has the certain property such that the desired something happens

3. The Construction Method: An Example If m and n are two consecutive integers, then 4 divides m 2 +n 2 -1. u A: m and n are two consecutive integers. u B: 4 divides m 2 +n 2 -1. The backward process produces B1: there exists an integer k such that m 2 +n 2 -1 = 4k. u Standard form: –Object: k –Certain Property: integer –Something Happens: m 2 +n 2 -1 = 4k.

4. Use A to construct and prove the desired object k: u A1: n = m +1. u A2: n 2 = (m+1) 2 = m 2 + 2m +1. u A3: m 2 +n 2 -1 = m 2 + m 2 + 2m +1 -1 = 2m 2 + 2m =2m(m+1) u A4: 2 divides m or 2 divides (m+1) u A5:2 divides m(m+1) Construct k = m(m+1)/2 u A6: k = m(m+1)/2 is an integer u A7: 4k = 4m(m+1)/2 = 2m(m+1) = m 2 +n 2 -1 u B1: There is an integer k such that m 2 +n 2 -1 = 4k.

5. The Choose Method u Use when the goal is a “for all” statement u Write the “for all” in standard form: B: For all objects with a certain property, something happens »Might proceed by making a list of all objects with the certain property and, for each, prove that the something happens. --- ok, if the list is not too long. u Usually we use the Choose Method which is to prove the statement: If x is an object with a certain property, then something happens.

6. The Choose Method: An Example If A  B = A, then A  B. The Backward Process (using definition) yields: B1: For all x in A, x is in B. Standard Form: –Object: x –Certain Property: in A –Something Happens: x is in B. u Choose Method: B2: If x in A, then x is in B.

7. Choose Example continued Prove: If A  B = A and if x is in A, then x is in B No reasonable key question so begin forward method.  A`: A  B = A and x is in A.  A`1: x is in A  B. u A`2: x is in A and x is in B u B`: x is in B.

8. Mathematical Induction u Use when the goal contains “for all integer n>0” u Write the “for all n” in standard form: B: For all integer n > 0, something happens Method: u Prove the something happens for n = 1. u Prove If something happens for n, then something happens for n+1.

9. Existence Method u Use when a known statement contains “there exist” u Write the “there exist” in standard form: A: There exists an object with a certain property such that something happens u Often we begin with the Forward Process assuming the existence of an object x with a certain property such that something happens.

10. Specialization Method u Use when a known statement contains “for all” u Write the “for all” in standard form: A: For all objects with a certain property, something happens u IDEA - When in the proof we discover an object x with the certain property, we may use A: to specialize for x and conclude that for x something happens.

11. Specialization Method: An Example If A  B, then A  B = A. u Forward Process yields from the hypothesis A1: For all x  A  x  B u Backward Process yields B1: A  A  B and A  B  A  B2: For all x  A  x  A  B and for all x  A  B  x  A  u The “and” in B2 indicates we have two sub-goals to prove: B2a: For all x  A  x  A  B. B2b: For all x  A  B  x  A 

12. u Use choose method to prove B2a.  Let a  A. -- ie, Prove If A1: and a  A, then B2a. u Now use specialization on the a. We have a  A and we thus specialize A1 for this a and obtain a  B. Thus a  A  B., which proves B2a. u Use choose method to prove B2b.  Let a  A  B. Note that implies that a  A which proves B2b.