=============================================================== Lect 9 - October 23, 2008 - ECS 20 - Fall 2008 - Phil Rogaway =============================================================== Today: o Quiz 2 o Relations o Functions - SCRIBE (none today, use your own notes for the class notes) ------------------------------------------ 1. Review of relations ------------------------------------------ DEF: A and B sets. Then a *relation* R is subset of A x B. R \subseteq A x B Defined last time: inverse of a relation composition of a relation Looked at: a graphical interpretation of relations ------------------------------------------ 2. Properties of Relations ------------------------------------------ Reflexive: x R x for all x Symmetric: x R y -> y R x for all x,y Transitive: x R y and y R x -> x R z for all x, y, z if R has all three properties, R is said to be an equivalence relation Make a table.... Reflexive Symmetric Transitive comments = on Integers Yes Yes Yes (or anything else) <= , integers Yes No Yes actually _antisymmetric_ Define this subseteq, sets Yes No Yes antisymmetric x E y if x and y are regular expressions and Yes Yes Yes blocks are the regular L(x) = L(y) languages x S y if x is a substring Yes No Yes of y x R y where x and y are strings and M is a some DFA and you go to the Yes Yes Yes same state on processing x and y x | y if 3 | x-y Yes Yes Yes Carefully prove this one and write out its blocks. Define when n | m ------------------------------------------ 3. Equivalence classes, quotients ------------------------------------------ Important definition: If R is an equivalence relation on A x A then [x] denotes the set of all elements related to x: [x] = { x': x R x'} We call [x] the *equivalence class* (or *block*) of x. The set of all equivalence classes of A with respect to a relation R is denoted A/R "quotient set of A by R", or "A mod R". I claim that ever equivalence relation *partitions* the universe into its blocks? What does this mean? Define a partitioning of the set A: Def: {A_i: i in I} is a partition of A if each A_i is nonempty, their union is A, and they are pairwise disjoint. Proposition: Let R be an equivalence relation on a set A. Then the blocks of R are a partition of A. Proof: didn't get to it. NOTATION: A/R the blocks of A relative to equivalence relation R. Note: you can talk about the blocks being related to one another by R, [x] R [y] iff x R y. Well-defined. Now go back to prior examples and identify the blocks in each case. When introduced partition, a student asked a question of we had earlier used our *tiles* to partition the (upper right quadrant) of the plane. I indicated that we had, but that one needs to be careful at the "edges" of each tile to make sure that each point is in only one tile. We defined [a, b) = {x\in\R a <= x < b) So a tile with left endpoint at (i,j) is is [i, i+1) x [j, j+1) and the plane is the disjoint union of tiles T_ij = [i, i+1) x [j, j+1) when i,j \in \N