Page 184 - HOW TO PROVE IT: A Structured Approach, Second Edition

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170 Relations
L(x, y, z) would be the set {(p, c, n) ∈ P × C × N | the person p has lived in
the city c for n years}.

Exercises

1. What are the truth sets of the following statements? List a few elements
∗
of each truth set.
(a) “x is a parent of y,” where x and y both range over the set P of all
people.
(b) “Thereissomeonewholivesin x andattends y,”where x rangesover
the set C of all cities and y ranges over the set U of all universities.
2. What are the truth sets of the following statements? List a few elements
of each truth set.
(a) “x lives in y,” where x ranges over the set P of all people and y
ranges over the set C of all cities.
(b) “The population of x is y,” where x ranges over the set C of all
cities and y ranges over N.
2
3. The truth sets of the following statements are subsets of R . List a few
elements of each truth set. Draw a picture showing all the points in the
plane whose coordinates are in the truth set.
2
(a) y = x − x − 2.
(b) y < x.
2
(c) Either y = x − x − 2or y = 3x − 2.
2
(d) y < x, and either y = x − x − 2or y = 3x − 2.
∗ 4. Let A ={1, 2, 3}, B ={1, 4}, C ={3, 4}, and D ={5}. Compute all
the sets mentioned in Theorem 4.1.3 and verify that all parts of the
theorem are true.
5. Prove parts 2 and 3 of Theorem 4.1.3.
∗
6. What’s wrong with the following proof that for any sets A, B, C, and
D,(A ∪ C) × (B ∪ D) ⊆ (A × B) ∪ (C × D)? (Note that this is the re-
verse of the inclusion in part 4 of Theorem 4.1.3.)

Proof. Suppose (x, y) ∈ (A ∪ C) × (B ∪ D). Then x ∈ A ∪ C and y ∈
B ∪ D, so either x ∈ A or x ∈ C, and either y ∈ B or y ∈ D. We con-
sider these cases separately.
Case 1. x ∈ A and y ∈ B. Then (x, y) ∈ A × B.
Case 2. x ∈ C and y ∈ D. Then (x, y) ∈ C × D.
Thus, either (x, y) ∈ A × B or (x, y) ∈ C × D,so(x, y) ∈ (A × B) ∪
(C × D).