Question

Consider the relation Rover the set N x N:(mi, ni)R(m2, 12) if mi + mzis even and ni + ng is even. Show R is an equivalence relation. Describe its equivalence classes.

Answer 1

Solutions cet R be the relator) on fet NXN Define (mi,no) Rama, na) if and on! only If miama & nitra is even, © Reflexire or for any em, n,) & AXN mitm, : 2m, seven nianian, even im,,n, Rom, n,) R is Reflexire ② symmetrics for cm,,ni) , (maina) E NXA (m,n) R (Ma, na 2 such that mitmaa eren & nitnaa even maam, ceren 4 natn, even (mo, nalrem, ino) R is symmetme ☺ Doansitive: For cm, mi), ema,na), (m 3,ng) EIR such (m.in, Rima,na) rma, na) R (M3,0g) thad

mitma seven nitna even marmaa eren * notna = even 2 mitmat mat mga even teren of nitrat 120g = even teren mitmatama= even & niang+ana=eren MiAma= even-ama of nitnaa erencana But ama fana is eren moamza eren nitna= even (m, n) R M 3, 3) R is Pransitive. ... R is equivalonce ted selation classin O tyreren en} Equiralance [cool] : { Cey) 110,00R (KY)} = {(x, y) ENXH4 ) 0+x= event = {(Xy) € MXN) * 4 y is even :[(0,0)] = {(min) E AVX AN l m an are even ce [(0,0)]={cam, and I minen [1] {(M) EN YN (9) R$)} [11,12)= {(my) ENXN/ are odd} [4,1)] {ldmat, 2011) mineN}, s & y

similiary [101] = {(219) I could R (Xiy)} {(x,y) 1 ge is eren 4 y is oudd} (10,1)] = { cam, anni) , minen} (1110)] = {CMy)(10) R (May)} [(110)] = {(14) l seis odd ay is even] [(1,0)]: {(2m-), an 21 mint } : Equilalance class are Ctbv)) [c , D)[Co 01 [Up 2).