Question

Prove equalities involving sets A, B, C and D a) (AIB)U(C1B) = (AUC) IB b) (AUB)-(ANB) = (A-8)U(-A) c) (AxB) OLC xD) - (ANC) x (BND) d) (AXB) (BAA) = (ANB)X(AMB)

Answer 1

• Definitions: Let M, N be two sets. - (1) MIN (OR M-N) is the set containing I all elements of M that are NOT in N. (2-) MUN is the set containing elements of M and n both. I common elements are • Written once.) - (3) MnN is the set of elements which are common in M and N. (4) MXN ={(m, n): mém, neng (5) QE MIN REM and a&N (6) 1 & MUN = 36 M + 3 + N ( be ) (+-) XEM AN . XEM and X EN - 18.) Two sets M and N are equal if for every EM X EN Scanned with CamScanner

(0) (818) u (CIB) = (AUC) 18 Let xe (AIB) u (C/B) =DE A1B or Q Ecle lor both) <=> OCCA and x&B or Xtc and x&B => XE A or a EC and x&B A XE AUC and x&B - XE (AUC) | B : (A1B) V. (C1B) = (AUC) 1B We can also use Venn diagram to show this (AWB) U(CIB) (AUC) IB : (AlB]uLc 18) = (AUC) 18 Scanned with CamScanner

(b) (AUB) - (ANB) = (A-B) U1B-A) + 1 + (4 U 8)-( 8 1 8) <=) XE AUB and a ¢ AAB X EA or X EB and x&A x&B & XEA and x&B or X EB and X & A e 1 & A - 3 6 3 6 8 - 1 # x € (A-B) U ( 6- A) : (AUB) - (ANB) = LA-B) U (8-A) Venn diagram diagram: LAUB) - (Ang) (A-B) U18-A) (-) Let ã € (A x B) n (CxD) Ī = (x,y) 12,4) E (AXB) n(CxD) Scanned with CamScanner

ES (Y) E AXB and (x, y) & cXD => XEA YEB and xtc, yed o X EA and XEC , y EB and yed ) « E(ANC), y E (BND) ♡ (,) E (ANC) x (BND) iã E (Anc) (BND) :. (AXB) n(C D) = (Anc) x (BND) (d.) Let (x, y) & (AXB) (BRA) E X EA YEB and x EB and yEA » X EA and x dB, Y EB and YEA - XEANBYE BnA = (x,y) & (ANB) x (BNA) : (AXB) n(Bnn) = (Ang) * (B1A) Scanned with CamScanner