18 2. Background (i) X = {1, 2} (ii) X = {1, 2, {1, 2}} (iii) X = {1, 2, {1, 3}} Exercise 2.2.4. Work inside the finite universe {1, 2,..., 10}. Define the following sets: A = {1, 3, 5, 7, 9} B = {1, 2, 3, 4, 5} C = {2, 4, 6, 8, 10} D = {7, 9} E = {4, 5, 6, 7} (i) Find all the subset relationships between pairs of the sets above. (ii) Which pairs, if any, are disjoint? (iii) Which pairs, if any, are complements? (iv) Find the following unions and intersections: A∪B, A∪D, B∩D, B ∩ E. We can also take unions and intersections of infinitely many sets. For sets Ai for i ∈ N, these are defined as follows. i Ai = {x : (∃i)(x ∈ Ai)} i Ai = {x : (∀i)(x ∈ Ai)} The i under the union or intersection symbol is also sometimes written “i ∈ N.” Exercise 2.2.5. For i ∈ N, let Ai = {0, 1,...,i} and let Bi = {0,i}. What are i Ai, i Bi, i Ai, and i Bi? If two sets are given by descriptions instead of explicit lists, we must prove one set is a subset of another by taking an arbitrary element of the first set and showing it is also a member of the second set. For example, to show the set of people eligible for President of the United States is a subset of the set of people over 30, we might say: Consider a person in the first set. That person must meet the criteria listed in the U.S. Constitution, which includes being at least

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