State Machines and Infinity 1 State diagrams state

State Machines and Infinity 1

State diagrams state transition action 2

Conway's Game of Life • Any live cell with fewer than two live neighbors dies, as if by underpopulation. • Any live cell with two or three live neighbors lives on to the next generation. • Any live cell with more than three live neighbors dies, as if by overpopulation. • Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction. • https: //playgameoflife. com/ 3

Glider Gun https: //en. wikipedia. org/wiki/Conway%27 s_ Game_of_Life 4

Countability Discrete Structures (CS 173) Adapted from Derek Hoiem, University of Illinois 5

Today’s class: countability • Are some infinite sets bigger than other infinite sets? • How big are these common infinite sets? – Naturals, integers, reals, rationals, powerset of naturals • What does it mean for a set to be “countable”? • How do we prove that a set is or is not countable? 6

Are there more integers than natural numbers? https: //www. youtube. com/watch? v=fa. QBr. AQ 87 l 4 https: //www. youtube. com/watch? v=Uj 3_Kqk. I 9 Zo 7

Are there more integers than natural numbers? • 8

Are there more rational numbers than integers? • 10

Countability A set is countably infinite if it has the same size as the set of natural numbers. A set is countable if it is finite or countably infinite. Countable sets: – Any subset of a countable set is countable – The Cartesian product of finitely many countable sets is countable – A union of countably many countable sets is countable 12

What sets are not countable? 13