LEt F: z rightarrow z be defined by F(n) = n + 5 Forall n elementof z let h: z r

Question

LEt F: z rightarrow z be defined by F(n) = n + 5 Forall n elementof z let h: z rightarrow z be defined by h(n) = 2h Forall n elementof z Find F(z_q) z_0 = set of odd integers Find h^-1(ze) z_e = set of even integers observe F oh= 9ze) z_e = set of even integers Observe F h: (n) rightarrow z is defined by (f h) (n) = F(h(n)) = F(2n) = 2n + 5 Forall n elementof z and h f: z rightarrow z is defined by (h F)(n) = h(F(n)) = h(n + 5) = 2 (n + 5) = 2n + 10 Forall n f. Find range of F h Find (range (F h)) Intersection(range (h f) For all the above pounce give reasons for your answer

Explanation / Answer

a) F(z0) = F (set of all odd integers ) = (set of all integers) + 5 = set of all even integers

b) H-1(ze) = H-1(set of all even integers)

where H (n) = 2n and H -1(n ) = n/2 for all valuers of n except n =0

H-1(set of all even integers) = set of all even integers / 2 (not equal to zero)

H-1(ze) = set of even integers except zero

c) Range of FoH i.e. 2n+5 where n is an integer

Range of FOH = set of all odd integers

d) Range of Hof = 2n+ 10 where n is an integer = set of all even integers

Intersection of range of FoH and range of HoF is null (not a single value)

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