Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 107 — #115
4.5. Finite Cardinality 107
(g) x D S z2y z.
This means that y is supposed to be a collection of sets, and x is the union of all of
them. A more concise notation for “ S z2y z’ is simply “S y.”
Class Problems
Problem 4.3.
Set Formulas and Propositional Formulas.
(a) Verify that the propositional formula .P AND Q/ OR .P AND Q/ is equivalent
to P .
(b) Prove that
for all sets, A; B, by showing
A D .A B/ [ .A \ B/
x 2 A IFF x 2 .A B/ [ .A \ B/
for all elements x using the equivalence of part (a) in a chain of IFF’s.
Problem 4.4.
Prove
Theorem (Distributivity of union over intersection).
A [ .B \ C / D .A [ B/ \ .A [ C / (4.8)
for all sets, A; B; C , by using a chain of iff’s to show that
x 2 A [ .B \ C / IFF x 2 .A [ B/ \ .A [ C /
for all elements x. You may assume the corresponding propositional equivalence 3.10.
Problem 4.5.
Prove De Morgan’s Law for set equality
A \ B D A [ B: (4.9)
by showing with a chain of IFF’s that x 2 the left-hand side of (4.9) iff x 2 the
right-hand side. You may assume the propositional version (3.14) of De Morgan’s
Law.