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

Question

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

Explanation / Answer

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 Ai is a subset of the prceding Ai - 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 = Nc = 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

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