Partition of a Set

Partition of a set Let S be a non empty set. A collection of non empty disjoint subsets of S whose union is S is called a partition of S. In other words, Let S be a non empty set (ie S≠$). A collection {A1, A2, AK} of subsets of S is Said to be a partition of S if (1) Ai, for each a Li AIAY, A2, Ax (11) Алајф, for ij [ie any two distinct sets in the collection are disjoint] [ie the sets in the collection are pairwize. disjoint And (iii) Ai = S [AQUALU VAL Example: Let S={1,2,3,4,5,6}. Let A1= {2,3,53, A2= {6}, As = {1,4}. Consider the collectim {A., Ar, As}. Here each Ai is non empty [ie Aid, A2, A34] NOW. AINAL= AinAj for izj ALO A,VALUA = {1,2,3,4,5,6}=s :EALALAS} is a partition of S. Let B, {1,3,53, BL = [2.4]. Consider the collection {B. Br]. Here B.UB= {1,2,24,5} #S Le BUB₂ #S {B1_Bu} is not a partition of s Consider the collection {{1,3}, {3,4}, {26}} Here {1,3}^{35} = {2} [1.33 73.53 $ .: This collection is not P partition of S. Noli The Subsets in the partition are blocks of the partition. called In the above example, A1. Az and Az are blocks L of the portition {A1, A2, A3} Result: Every equivalenie relation on a non-empty set induies a partition of that set. ie if R is an equivalence relation on a non-empty set A, then the equivalence Clasies of A form a partition of A. ve, the quotient set A/R is a partition of A. es Let A = {1,2,3,4,5,6] and R= {1,1), (15), (2, 2),(2,3),(2,6), (3,2), (3, 3), (3,6), (4,4), (5-1), (5,5),(6,2),(6,3),(6,6)} clearly R is an equivalenie relctim of A. -R induies a partition of A. ie, the equivalence classes of A form a Partition of A- ie A/R is partition of A. Now [] = {1.5} [2] = {2,3,6} [4] 2 £43. A/R = {[1][2][4]] To check A/R is Now [1][2] = Φ [1]~[4] =ф a partition of A [2] ^ [4] = Φ. [1], [2] and [4] are non empty sets. 2],[1] [ and [4] are pairwise disjoink- Also, [1]U[Q]U[4] = {1.5}u{2,3,6} U{4} ={1,2,3-4,5,6} = A :. A/R is Partition of A Noter Let R be an equivalene relation on a set A. let a EA We lenow that the equivalence class of a is denoted as [a] Esnivalente class of a is also denoted as Ria) Find all the equivalenie classes of the set {0,1,2,..., 10} under the relation a = b(mod 5) OR Find all the egnivalence classes of the set {0,1,2-10} under the relation congruenie modulo 5' Let A = {0,1,2,3--10} Now, [0] = {0,5,10} [1] = {1,6} [2] = २,٦ [3] = {3,8} [4] = {4,93 Thus the equivalence classes are [0], [1] [2], [3] and [4] ie esnivalence classes are {0.5,10}, {1.6}{0.7} {3,8} and {4.9} [9]={XEA: IRC} = {xEA: xza mods) Note: Consider the above problem. Here egnivdenie classes of A are [0]_[]_[2], [3] and [4]. Farm this we set a partition P of A. ve Partition, P = {[0][1][2][3],[49] le P= {(0,5,10}, {16}, (2.7}, {20} {4.93} Note: P is also woitten as AIR (Quotient set of AbyR). Thus, form the above problem, A/R = {[0] [1], [2], [3]_[4]} 2 Find all the equivalence classes on the set {0,1, 7} under the relation "Congruenie modulo 4' [ie a=b(mod 4)] Let A = 20,1,2,7 [a] = {xEA: xRa} = {xEA: x = a (mod4)} [0] = {0,4} Now, [1] = {1,53 [2] = {26} [3] 2 {3,7} Heze xRa means 1=(mod 4) Thus the equivalence classes are [0] [1][2] and [3] ie equivalenie classes are {0,4}, {1.5} {2,6} And 3,73 Noli In the above problem Partition, Por A/R = {[0]_[]_[2]_[]} = {{0.4}, {15}, {0,6}, {27}} 3 Let z be the set of integers and let R be the relation called congruenie modulo 3'. Determine the egnivelenie classes generated by the elements of 2 Z={------3-2-10,1,2,3,----- [a] = {xEZ: Ra} = {xcZ : x = a (mod 3)} Now, [0] = {-9-6-3,0,3,6,9,---} [1]={-,-8,-5,-2, 1.4,7,10,--} []={-7, -4, -1, 2,5,8,--} Thus, the equivalenie classes are [0] [1] and [2] Note: In the above problem, Partitim, P= {[0], [1][2]} 4 Let A={1,2,3,4} and P= {{1,2,3}, {4}} be a Partition of A. Find the equivalence relation determined by P Criven, P = {{1,2,3}, {41} Esnivdenne relation, R = {0,1), (1,2),(13),(2.1), (2,2),(2,3), (3,1), (32),(3,3),(4.4)} 5 If P= {{2}, {4}, {5,73} is a partition of a set A, find the equivalence relation determined by P. Given P = {{2}, {4}, {5,73} Esnivelenie relation, R = {(22), (4,4), (5,5),(6,7),(1.5),(7,7)} 6 Let A={1,2,3,4} and let R= {11),(2)_ (2,1), (2,2), (34), (4,3), (3, 3), (4,4)} be an esnivalee relation on A Determine the partition P. (ie determine A/R) R= {1,1), (1,2),(2,1),(2,2), (3,4),(4,3), (3, 3), (4,4)} Partihin, P= {{1,2}, {3,4}} 4 Let A={1,2,3,4} and P= {{1,2,3}, {4}} be a Partition of A. Find the equivalence relation determined by P Criven, P = {{1,2,3}, {41} Esnivdenne relation, R = {0,1), (1,2),(13),(2.1), (2,2),(2,3), (3,1), (32),(3,3),(4.4)} 5 If P= {{2}, {4}, {5,73} is a partition of a set A, find the equivalence relation determined by P. Given P = {{2}, {4}, {5,73} Esnivelenie relation, R = {(22), (4,4), (5,5),(6,7),(1.5),(7,7)} 6 Let A={1,2,3,4} and let R= {11),(2)_ (2,1), (2,2), (34), (4,3), (3, 3), (4,4)} be an esnivalee relation on A Determine the partition P. (ie determine A/R) R= {1,1), (1,2),(2,1),(2,2), (3,4),(4,3), (3, 3), (4,4)} Partihin, P= {{1,2}, {3,4}} ie aRb bra - Ris Symmetric. To prove Ris transitive Let ars and bre. é a = b(mod m) and b = c (mod m) =mla-b and m/b-c Then mla-b+b.c le mla-c a = c (mod m) The, a Rb and bRe aRe .. R is transitive. If milk, and mik, then m/kitk Sime Ris reflescive, Symmetne and transitive Ris an esnivalenie relation. 8 Determine whether the relation R represented by the matrix MR: 01 OL an equivalence relation. MR 01 [ D 1 2 1 1 D D & MR = 1 : R = {(1,1),(2,2) (2,3), (3, 2), (33)} Now, (1, 1) ER, (2, 2) ER and (33) ER - R is reflexive. Now, (2,3)ER. Hiro (3,2) ER ما R is symmetorn Now, (3,2) ER and (2,3) ER. Al10 (33) ER (32JER and (2,3) ER = (3,3) ER ... R is transitive Since R is reflexive symmetric and transitive, R is an equivalence relation.