Please help me with the following I do not understand how to

Please help me with the following I do not understand how to do this. Thank you!

For each natural number n, let A_n = {k element N|k greaterthanorequalto n} and U = N. Write down the following sets: Union^5 _j = 1 A_j (Intersection^5 _j = 1 A_j)^c Intersection^5 _j = 1 A^c _j Union_j element N A_j For each r element (0, infinity), define T_r = [-r^2, r^2]. Determine each of the following sets: Intersection_r element (0, infinity) T_r Union_r element (0, infinity) T_r Intersection_k element N T_k Union_k element N T_k| For each n element N, let S_n = [0, 1/n] times [0, 1/n]. What are Intersection_n element N S_n and Union_n element N S_n

Solution

Preparatory work common to all parts of Q1

A₁ = {1, 2, 3, 4, 5, 6, 7, ........} = N

A₂ = {2, 3, 4, 5, 6, 7, ........} = N - {1}

A₃ = {3, 4, 5, 6, 7, ........} = N - {1, 2}

A₄ = {4, 5, 6, 7, ........} = N - {1, 2, 3}

A₅ = {5, 6, 7, ........} = N - {1, 2, 3, 4}

Also note that each A_i is a subset of the prceding A_{i - 1}

Now,

Solution for 1(a)

Union of A₁, A₂, A₃, A₄and A₅ is the largest of the five sets, namely A₁ = N

Solution for 1(b)

Intersection of A₁, A₂, A₃, A₄ and A₅ is the smallest of the five sets, namely A₅ = N - {1, 2, 3, 4}

Solution for 1(c) [Note that given the universal set U is N]

Complement of A₁ = {1, 2, 3, 4, 5, 6, 7, ........}^c = N^c = Null set   

Complement of A₂ = {2, 3, 4, 5, 6, 7, ........}^c = [N - {1}]^c = {1}

Complement of A₃ = {3, 4, 5, 6, 7, ........}^c = [N - {1, 2}]^c = {1, 2}

Complement of A₄ = {4, 5, 6, 7, ........}^c = [N - {1, 2, 3}]^c = {1, 2, 3}

Complement of A₅ = {5, 6, 7, ........}^c = [N - {1, 2, 3, 4}]^c = {1, 2, 3, 4}

Clearly, intersection of the above complementary sets is the null set.

Solution for 1(d)

Union of A₁, A₂, A₃, A₄ ....... is clearly the largest set namely, A₁ ANSWER