Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 109 — #117
4.5. Finite Cardinality 109
Homework Problems
Problem 4.8.
Let A, B and C be sets. Prove that
A [ B [ C D .A B/ [ .B C / [ .C A/ [ .A \ B \ C / (4.10)
using a chain of IFF’s as Section 4.1.5.
Problem 4.9.
Union distributes over the intersection of two sets:
A [ .B \ C / D .A [ B/ \ .A [ C / (4.11)
(see Problem 4.4).
Use (4.11) and the Well Ordering Principle to prove the Distributive Law of
union over the intersection of n sets:
A [ .B 1 \ \ B n 1 \ B n /
D .A [ B 1 / \ \ .A [ B n 1 / \ .A [ B n / (4.12)
Extending formulas to an arbitrary number of terms is a common (if mundane)
application of the WOP.
Exam Problems
Problem 4.10.
You’ve seen how certain set identities follow from corresponding propositional
equivalences. For example, you proved by a chain of iff’s that
.A B/ [ .A \ B/ D A
using the fact that the propositional formula .P ANDQ/OR.P ANDQ/ is equivalent
to P .
State a similar propositional equivalence that would justify the key step in a proof
for the following set equality organized as a chain of iff’s:
A B D A C [ .B \ C / [ A [ B \ C (4.13)
(You are not being asked to write out an iff-proof of the equality or to write out
a proof of the propositional equivalence. Just state the equivalence.)