Page 96 - HOW TO PROVE IT: A Structured Approach, Second Edition

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82 Quantificational Logic
For each y ∈ Y, let A y ={p ∈ P | the person p was alive at some time
during the year y}. Find ∪ y∈Y A y and ∩ y∈Y A y .
∗ 8. Let I ={2, 3}, and for each i ∈ I let A i ={i, 2i} and B i ={i, i + 1}.
(a) List the elements of the sets A i and B i for i ∈ I.
(b) Find ∩ i∈I (A i ∪ B i ) and (∩ i∈I A i ) ∪ (∩ i∈I B i ). Are they the same?
(c) In parts (c) and (d) of exercise 2 you analyzed the statements x ∈
∩ i∈I (A i ∪ B i ) and x ∈ (∩ i∈I A i ) ∪ (∩ i∈I B i ). What can you conclude
from your answer to part (b) about whether or not these statements
are equivalent?
9. Give an example of an index set I and indexed families of sets {A i | i ∈ I}
and {B i | i ∈ I} such that ∪ i∈I (A i ∩ B i ) = (∪ i∈I A i ) ∩ (∪ i∈I B i ).
10. ShowthatforanysetsAand B, P (A ∩ B) = P (A) ∩ P (B),byshowing
that the statements x ∈ P (A ∩ B) and x ∈ P (A) ∩ P (B) are equiva-
lent. (See Example 2.3.3.)
11. Give examples of sets A and B for which P (A ∪ B) = P (A) ∪ P (B).
∗
12. Verify the following identities by writing out (using logical symbols)
what it means for an object x to be an element of each set and then using
logical equivalences.
(a) ∪ i∈I (A i ∪ B i ) = (∪ i∈I A i ) ∪ (∪ i∈I B i ).
(b) (∩F) ∩ (∩G) =∩(F ∪ G).
(c) ∩ i∈I (A i \ B i ) = (∩ i∈I A i ) \ (∪ i∈I B i ).
13. Sometimes each set in an indexed family of sets has two indices. For
∗
this problem, use the following definitions: I ={1, 2}, J ={3, 4}.For
each i ∈ I and j ∈ J, let A i, j ={i, j, i + j}. Thus, for example, A 2,3 =
{2, 3, 5}.
(a) For each j ∈ J let B j =∪ i∈I A i, j = A 1, j ∪ A 2, j . Find B 3 and B 4 .
(b) Find ∩ j∈J B j . (Note that, replacing B j with its definition, we could
say that ∩ j∈J B j =∩ j∈J (∪ i∈I A i, j ).)
(c) Find ∪ i∈I (∩ j∈J A i, j ). (Hint: You may want to do this in two
steps, corresponding to parts (a) and (b).) Are ∩ j∈J (∪ i∈I A i, j ) and
∪ i∈I (∩ j∈J A i, j ) equal?
(d) Analyze the logical forms of the statements x ∈∩ j∈J (∪ i∈I A i, j ) and
x ∈∪ i∈I (∩ j∈J A i, j ). Are they equivalent?
14. (a) Show that if F = ∅, then the statement x ∈∪F will be false no
matter what x is. It follows that ∪∅ = ∅.
(b) Show that if F = ∅, then the statement x ∈∩F will be true no
matter what x is. In a context in which it is clear what the universe
of discourse U is, we might therefore want to say that ∩∅ = U.
However, this has the unfortunate consequence that the notation ∩∅
will mean different things in different contexts. Furthermore, when