Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 106 — #114
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Chapter 4
Mathematical Data Types
Problems for Section 4.1
Practice Problems
Problem 4.1.
For any set A, let pow.A/ be its power set, the set of all its subsets; note that A is
itself a member of pow.A/. Let ; denote the empty set.
(a) The elements of pow.f1; 2g/ are:
(b) The elements of pow.f;; f;gg/ are:
(c) How many elements are there in pow.f1; 2; : : : ; 8g/?
Problem 4.2.
Express each of the following assertions about sets by a formula of set theory. 7
Expressions may use abbreviations introduced earlier (so it is now legal to use “D”
because we just defined it).
(a) x D ;.
(b) x D fy; zg.
(c) x y. (x is a subset of y that might equal y.)
Now we can explain how to express “x is a proper subset of y” as a set theory
formula using things we already know how to express. Namely, letting “x ¤ y”
abbreviate NOT.x D y/, the expression
.x y AND x ¤ y/;
describes a formula of set theory that means x y.
From here on, feel free to use any previously expressed property in describing
formulas for the following:
(d) x D y [ z.
(e) x D y z.
(f) x D pow.y/.
7 See Section 8.3.2.