Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 116 — #124
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Chapter 4
Mathematical Data Types
Problem 4.27.
Let R be a binary relation on a set D. Let x; y be variables ranging over D. Indicate
the expressions below whose meaning is that R is an injection Π1 in. Remember
R is a not necessarily total or a function.
1. R.x/ D R.y/ IMPLIES x D y
2. R.x/ \ R.y/ D ; IMPLIES x ¤ y
3. R.x/ \ R.y/ ¤ ; IMPLIES x ¤ y
4. R.x/ \ R.y/ ¤ ; IMPLIES x D y
5. R 1 .R.x// D fxg
6. R 1 .R.x// fxg
7. R 1 .R.x// fxg
8. R.R 1 .x// D x
Problem 4.28.
The language of sets and relations may seem remote from the practical world of
programming, but in fact there is a close connection to relational databases, a
very popular software application building block implemented by such software
packages as MySQL. This problem explores the connection by considering how to
manipulate and analyze a large data set using operators over sets and relations. Systems
like MySQL are able to execute very similar high-level instructions efficiently
on standard computer hardware, which helps programmers focus on high-level design.
Consider a basic Web search engine, which stores information on Web pages and
processes queries to find pages satisfying conditions provided by users. At a high
level, we can formalize the key information as:
A set P of pages that the search engine knows about
A binary relation L (for link) over pages, defined such that p 1 L p 2 iff page
p 1 links to p 2
A set E of endorsers, people who have recorded their opinions about which
pages are high-quality