Chapter 1Solutions to Selected ExercisesSection 1.12.{2,4}3.{7,10}5.{2,3,5,6,8,9}6.{1,3,5,7,9,10}8.A9.∅11.B12.{1,4}14.{1}15.{2,3,4,5,6,7,8,9,10}18. 119. 322. We find thatB={2,3}. SinceAandBhave the same elements, they are equal.23. Letx∈A. Thenx= 1,2,3. Ifx= 1, since 1∈Z+and 12<10, thenx∈B. Ifx= 2, since2∈Z+and 22<10, thenx∈B. Ifx= 3, since 3∈Z+and 32<10, thenx∈B. Thus ifx∈A, thenx∈B.Now suppose thatx∈B. Thenx∈Z+andx2<10. Ifx≥4, thenx2>10 and, for thesevalues ofx,x /∈B. Thereforex= 1,2,3. For each of these values,x2<10 andxis indeed inB. Also, for each of the valuesx= 1,2,3,x∈A. Thus ifx∈B, thenx∈A. ThereforeA=B.26. Since (−1)3−2(−1)2−(−1) + 2 = 0,−1∈B. Since−1/∈A,A=B.27. Since 32−1>3, 3/∈B. Since 3∈A,A=B.30. Equal31. Not equal34. Letx∈A. Thenx= 1,2. Ifx= 1,x3−6x2+ 11x= 13−6·12+ 11·1 = 6.Thusx∈B. Ifx= 2,x3−6x2+ 11x= 23−6·22+ 11·2 = 6.Againx∈B. ThereforeA⊆B.35. Letx∈A. Thenx= (1,1) orx= (1,2). In either case,x∈B. ThereforeA⊆B.38. Since (−1)3−2(−1)2−(−1) + 2 = 0,−1∈A. However,−1/∈B. ThereforeAis not a subsetofB.39. Consider 4, which is inA. If 4∈B, then 4∈Aand 4 +m= 8 for somem∈C. However, theonly value ofmfor which 4 +m= 8 ism= 4 and 4/∈C. Therefore 4/∈B. Since 4∈Aand4/∈B,Ais not a subset ofB.1Preview Mode
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