Sets & Functions. x A. 7 Z, 2/3 Z, Z pow(r) Examples: {7, Albert R., π/2, T} x is a member of A: x A
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1 Mathematics for Computer Science MIT 6.042J/18.062J Sets & Functions What is a Set? Informally: set is a collection of mathematical objects, with the collection treated as a single mathematical object. (This is circular of course: what s a collection?) lec 2F.1 lec 2F.2 Some sets real numbers, R complex numbers, C integers, Z empty set, set of all subsets of integers, pow(z) the power set Some sets {7, lbert R., π/2, T} set with 4 elements: two numbers, a string, and a oolean. Same as {T, lbert R., 7, π/2} -- order doesn t matter lec 2F.3 lec 2F.4 Membership x is a member of : x π/2 {7, lbert R., π/2, T} π/3 {7, lbert R., π/2, T} 14/2 {7, lbert R., π/2, T} Synonyms for Membership x x is an element of x is in Examples: 7 Z, 2/3 Z, Z pow(r) lec 2F.5 lec 2F.6 1
2 In or Not In n element is in or not in a set: {7, π/2, 7} is same as {7, π/2} (No notion of being in the set more than once) Subset ( ) is a subset of is contained in Every element of is also an element of : x [x IMPLIES x ] lec 2F.7 lec 2F.8 examples: Subset Z R, R C, {3} {5,7,3}, every set everything def: x [x IMPLIES x ] false true lec 2F.9 lec 2F.10 New sets from old union Venn Diagram for 2 Sets lec 2F.14 lec 2F.15 2
3 intersection set difference lec 2F.16 lec 2F.17 set-theoretic equality set-theoretic equality ( C) = ( ) ( C) proof: Show these have the same elements, namely, x Left Hand Set iff x RHS for all x. ( C) = ( ) ( C) proof uses fact from last time: P OR (Q ND R) equiv (P OR Q) ND (P OR R) lec 2F.18 lec 2F.19 set-theoretic equality set-theoretic equality ( C) = ( ) ( C) proof: x ( C) x OR x ( C) iff (def of ) iff x OR (x ND x C) (def ) iff (x OR x ) ND (x OR x C) (by the equivalence) proof: (x OR x )ND(x OR x C) iff (x )ND(x C) (def ) iff x ( ) ( C) (def ). QED lec 2F.20 lec 2F.21 3
4 is taking subject relation Relations & Functions students is taking subjects lec 2F.25 Feb 17 Image by MIT OpenCourseWare. Copyright lec 2F.26 formula evaluation relation arithmetic formulas numbers evaluates to nonstop bus trip relation cities cities nonstop bus oston oston sqrt(9) Providence Providence 50/ New York New York Feb 17 Copyright lec 2F.27 lec 2F.28 inary relations binary relation, R, from a set to a set associates of elements of with elements of. inary relation R from to domain R: codomain b 1 a 1 b 2 a 2 b 3 a 3 arrows b 4 lec 2F.33 Feb 17 lec 2F.34 4
5 inary relation R from to domain R: codomain b 1 inary relation R from to R: b 1 a 1 b 2 a 1 b 2 a 2 b 3 a 2 b 3 a 3 b 4 Feb. 17, a 3 graph(r) ::= the arrows b 4 lec 2F.35 Feb. 17, graph(r) = {(a 1,b 2 ), (a 1,b 4 ), (a 3,b 4 )} lec 2F.37 archery on relations,, = 1 arrow out,,= 1 arrow in f: function, f, from to is a relation which associates each element, a, of with at most one element of. called f(a) Feb. 17, 2009 lec 2F.38 Feb. 17, lec 2F.39 function archery function archery 1 arrow out 1 arrow out Feb. 17, 2009 lec 2F.40 Feb. 17, 2009 lec 2F.41 5
6 function archery 1 arrow out f( ) = total relations R: is total iff every element of is associated with at least one element of Feb. 17, 2009 lec 2F.42 Feb. 17, 2009 lec 2F.44 total relation archery total relation archery 1 arrow out 1 arrow out Feb. 17, 2009 lec 2F.45 Feb. 17, 2009 lec 2F.46 1 arrow out total relation archery total & function archery exactly 1 arrow out f( ) = Feb. 17, 2009 lec 2F.47 Feb. 17, 2009 lec 2F.49 6
7 surjections (onto) R: is a surjection iff every element of is associated with some element of surjection archery 1 arrow in Feb. 17, 2009 lec 2F.53 Feb. 17, 2009 lec 2F.54 surjection archery surjection archery 1 arrow in 1 arrow in Feb. 17, 2009 lec 2F.55 Feb. 17, 2009 lec 2F.56 surjective & function injection archery 1 arrow out 1 arrow in 1 arrow in Feb. 17, 2009 lec 2F.58 Feb. 17, 2009 lec 2F.62 7
8 injection archery injection archery 1 arrow in 1 arrow in Feb. 17, 2009 lec 2F.63 Feb. 17, 2009 lec 2F.64 bijection archery exactly 1 arrow out exactly 1 arrow in Team Problems Problems 1 3 Feb. 17, 2009 Copyright lbert R. February Meyer, 12, ll rights reserved. lec 2F.69 lec 2F.71 8
9 MIT OpenCourseWare J / J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit:
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