[tex]A=\left \{ 0,1,2,3,4,5\right \}[/tex]
[tex]a)B=\left \{ x\:|\:x=a+3\: \: si \: \:a\:\in\:\mathbb{A}\right \}[/tex]
[tex]x = a + 3[/tex]
[tex]pt. \: a = 0 = > x = 0 + 3 = 3[/tex]
[tex]pt. \: a = 1 = > x = 1 + 3 = 4[/tex]
[tex]pt. \: a = 2 = > x = 2 + 3 = 5[/tex]
[tex]pt. \: a = 3= > x = 3 + 3 = 6[/tex]
[tex]pt. \: a= 4 = > x = 4 + 3 = 7[/tex]
[tex]pt. \: a = 5 = > x = 5 + 3 = 8[/tex]
[tex] = > B=\left \{ 3,4,5,6,7,8 \right \}[/tex]
[tex]b)C=\left \{ y\:|\:y=3^{a}-1\:si\:a\:\in\:\mathbb{A}\right \}[/tex]
[tex]y = {3}^{a} - 1[/tex]
[tex]pt. \: a = 0 = > y = {3}^{0} - 1 = 1 - 1 = 0[/tex]
[tex]pt. \: a = 1 = > y = {3}^{1} - 1 = 3 - 1 = 2[/tex]
[tex]pt. \: a = 2 = > y = {3}^{2} - 1 = 9 - 1 = 8[/tex]
[tex]pt. \: a = 3 = > y={3}^{3} - 1 = 27 - 1 = 26[/tex]
[tex]pt. \: a = 4 = > y = {3}^{4} - 1 = 81 - 1 = 80[/tex]
[tex]pt. \: a = 5 = > y = {3}^{5} - 1 = 243 - 1 = 242[/tex]
[tex] = > C=\left \{ 0,2,8,26,80,242\right \}[/tex]
[tex]c)D=\left \{ x\:|\:x =\overline{5a}\: \: si \: \:a\:\in\:\mathbb{A}\right \}[/tex]
[tex]pt. \: a = 0 = > x = 50[/tex]
[tex]pt. \: a = 1 = > x = 51[/tex]
[tex]pt. \: a = 2 = > x = 52[/tex]
[tex]pt. \: a = 3 = > x = 53[/tex]
[tex]pt. \: a = 4 = > x = 54[/tex]
[tex]pt. \: a = 5 = > x = 55[/tex]
[tex] = > D=\left \{ 50,51,52,53,54,55 \right \}[/tex]
[tex]d)E=\left \{ y\:|\:y =3^{a}-2^{a} \: \:si \: \:a\:\in\:\mathbb{A}\right \}[/tex]
[tex]y = {3}^{a} - {2}^{a} [/tex]
[tex]pt. \: a = 0 = > y = {3}^{0} - {2}^{0} = 1 - 1 = 0[/tex]
[tex]pt. \: a = 1 = > y={3}^{1} - {2}^{1} = 3 - 2 = 1[/tex]
[tex]pt. \: a = 2 = > y={3}^{2} - {2}^{2} = 9 - 4 = 5[/tex]
[tex]pt. \: a = 3 = > y = {3}^{3} - {2}^{3} = 27 - 8 = 19[/tex]
[tex]pt. \: a = 4 = > y={3}^{4} - {2}^{4} = 81 - 16 = 65[/tex]
[tex]pt. \: a = 5 = > y = {3}^{5} - {2}^{5} = 243 - 32 = 211[/tex]
[tex] = > E=\left \{ 0,1,5,19,65,211 \right \}[/tex]
[tex]e)F=\left \{ x\:|\:x =\overline{aa}\: \: si \: \:a\:\in\:\mathbb{A}\right \}[/tex]
[tex]x = \overline{aa}[/tex]
[tex]pt. \: a = 0 = > x \: nu \: exista[/tex]
[tex]pt. \: a = 1 = > x = 11[/tex]
[tex]pt. \: a = 2 = > x = 22[/tex]
[tex]pt. \: a = 3 = > x = 33[/tex]
[tex]pt. \: a = 4= > x = 44[/tex]
[tex]pt. \: a = 5 = > x = 55[/tex]
[tex] = > F=\left \{ 11,22,33,44,55 \right \}[/tex]
[tex]f)G=\left \{ x\:|\:x =2a^{2}-2\: \: si \: \:a\:\in\:\mathbb{A}\right \}[/tex]
[tex]x = 2 {a}^{2} - 2[/tex]
[tex]pt. \: a = 0 = > x = 2 \times {0}^{2} - 2 = 2 \times 0 - 2 = 0 - 2 = - 2[/tex]
[tex]pt. \: a = 1 = > x = 2 \times {1}^{2} - 2 = 2 \times 1 - 2 = 2 - 2 = 0[/tex]
[tex]pt. \: a = 2 = > x=2 \times {2}^{2} - 2 = 2 \times 4 - 2 = 8 - 2 = 6[/tex]
[tex]pt. \: a = 3 = > x = 2 \times {3}^{2} - 2 = 2 \times 9 - 2 = 18 - 2 = 16[/tex]
[tex]pt. \: a = 4 = > x=2 \times {4}^{2} - 2 = 2 \times 16 - 2 = 32 - 2 = 30[/tex]
[tex]pt. \: a = 5 = > x = 2 \times {5}^{2} - 2 = 2 \times 25 - 2 = 50 - 2 = 48[/tex]
[tex] = > G=\left \{ -2,0,6,16,30,48\right \}[/tex]
