Page 95 - HOW TO PROVE IT: A Structured Approach, Second Edition

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More Operations on Sets 81
distributive law for universal quantification and conjunction to show that
the elementhood tests for the two sets are equivalent.
7. ∪ b∈B L b ={t |∃b ∈ B(t ∈ L b )}={t ∈ S |∃ b(b ∈ B ∧ L(b, t))}. But
B was defined to be the set of all students who are admired by all
students, so b ∈ B means b ∈ S ∧∀s ∈ SA(s, b). Inserting this, we get
∪ b∈B L b ={t ∈ S |∃b(b ∈ S ∧∀s ∈ SA(s, b) ∧ L(b, t))}= the set of all
students who are liked by some student who is admired by all students.

Exercises

∗ 1. Analyze the logical forms of the following statements. You may use the
symbols ∈, /∈, =, =, ∧, ∨, →, ↔, ∀, and ∃ in your answers, but not
⊆, ⊆, P , ∩, ∪, \, {, },or ¬. (Thus, you must write out the definitions
of some set theory notation, and you must use equivalences to get rid of
any occurrences of ¬.)
(a) F ⊆ P (A).
(b) A ⊆{2n + 1 | n ∈ N}.
2
(c) {n + n + 1 | n ∈ N}⊆{2n + 1 | n ∈ N}.
(d) P (∪ i∈I A i ) ⊆∪ i∈I P (A i ).
2. Analyze the logical forms of the following statements. You may use the
symbols ∈, /∈, =, =, ∧, ∨, →, ↔, ∀, and ∃ in your answers, but not
⊆, ⊆, P , ∩, ∪, \, {, },or ¬. (Thus, you must write out the definitions
of some set theory notation, and you must use equivalences to get rid of
any occurrences of ¬.)
(a) x ∈∪F \∪G.
(b) {x ∈ B | x /∈ C}∈ P (A).
(c) x ∈∩ i∈I (A i ∪ B i ).
(d) x ∈ (∩ i∈I A i ) ∪ (∩ i∈I B i ).
3. We’ve seen that P (∅) = {∅}, and {∅} = ∅. What is P ({∅})?
4. Suppose F = {{red, green, blue}, {orange, red, blue}, {purple, red,
∗
green, blue}}. Find ∩F and ∪F.
5. Suppose F ={{3, 7, 12}, {5, 7, 16}, {5, 12, 23}}. Find ∩F and ∪F.
6. Let I ={2, 3, 4, 5}, and for each i ∈ I let A i ={i, i + 1, i − 1, 2i}.
(a) List the elements of all the sets A i , for i ∈ I.
(b) Find ∩ i∈I A i and ∪ i∈I A i .
7. Let P ={Johann Sebastian Bach, Napoleon Bonaparte, Johann
Wolfgang von Goethe, David Hume, Wolfgang Amadeus Mozart, Isaac
Newton, George Washington} and let Y ={1750, 1751, 1752, . . . , 1759}.