Σ 2i = 2 k i=0 CSC 225: Proof of the Day

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Presentation transcript:

Σ 2i = 2 k+1 - 1 i=0 CSC 225: Proof of the Day Prove by induction that: k Σ 2i = 2 k+1 - 1 i=0 Put your name on your answer and hand in your proof. All attempts will be given full participation marks (correct or not).

Announcements Note: The midterm is scheduled on Wed. Oct. 26 so that I can hand it back marked before the drop deadline (Oct. 31). Any questions about the course outline? Assignment #1 and Tutorial #1 are posted. Tutorials start next week. Bring your schedule to class on Tuesday (to help me in choosing office hours).

Duncan Hogg (dshogg@uvic.ca), ECS 230 Computer Science COOP Application deadline: Thursday Sept. 15 Application form available outside the ECSM Co-op Office (ECS 204). To learn more about the Co-operative Education Program and and Career Services on campus, visit: http://www.uvic.ca/coopandcareer OR see/email: Duncan Hogg (dshogg@uvic.ca), ECS 230 Computer Science also offers a work experience program for students wanting the benefits of work experience but not a full COOP program.

Hamilton Cycles A cycle which includes all the vertices of a graph. Fullerenes: 3-regular planar graphs, face sizes 5 and 6. Conjecture: Every fullerene has at least one Hamilton cycle.

The Petersen graph has no Hamilton cycles.

Review of Induction

Overview Questions from last class Review of induction Induction is very similar to recursion and one goal of this class is to become skilled at writing recursive programs. It is a useful tool for proving that programs are correct. Time complexities of algorithms will be computed by solving recurrence relations. Induction can then be used to prove that the answers you find are correct.

Natural Numbers = { 0, 1, 2, 3, 4, … } Inductive Definition: [Basis] 0 is in the set [Inductive step]: If k is in then k+1 is in

Complete Binary Trees: Height 0 Height 1 Height 2 Height 3

root vertex Leaves: vertices with one incident edge Height: maximum distance from root to a leaf measured by number of edges on the path.

Height 0 Height 1 Height 1 Height 2 Height 2 Height 3

Height 0 Height 1 Height 1 Height 2 Height 2 Height 3

r r1 r2

How can we give an inductive definition of a complete binary tree of height h? How many nodes does a complete binary tree of height h have? Prove the answer is correct by induction. Create a recurrence relation T(h) where T(h) gives the number of nodes of a complete binary tree of height h. Solve your recurrence relation and prove the answer is correct by induction.