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