Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 486 — #494
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Chapter 12
Simple Graphs
Problem 12.15.
Because of the incredible popularity of his class Math for Computer Science, TA
Mike decides to give up on regular office hours. Instead, he arranges for each
student to join some study groups. Each group must choose a representative to talk
to the staff, but there is a staff rule that a student can only represent one group. The
problem is to find a representative from each group while obeying the staff rule.
(a) Explain how to model the delegate selection problem as a bipartite matching
problem. (This is a modeling problem; we aren’t looking for a description of an
algorithm to solve the problem.)
(b) The staff’s records show that each student is a member of at most 4 groups,
and all the groups have 4 or more members. Is that enough to guarantee there is a
proper delegate selection? Explain.
Problem 12.16.
Let bR be the “implies” binary relation on propositional formulas defined by the rule
that
F bR G iff Œ.F IMPLIES G/ is a valid formula: (12.4)
For example, .P AND Q/ bR P , because the formula .P AND Q/ IMPLIES P is
valid. Also, it is not true that .P OR Q/ bR P since .P OR Q/ IMPLIES P is not
valid.
(a) Let A and B be the sets of formulas listed below. Explain why bR is not a weak
partial order on the set A [ B.
(b) Fill in the bR arrows from A to B.