Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 84 — #92
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Chapter 3
Logical Formulas
If a function is differentiable, then it is continuous
is true, as indeed it is.
But wait a minute! The implication
If a function is differentiable, then it is not continuous
is completely false. So we could conclude that its converse
If a function is not continuous, then it is differentiable,
should be true, but in fact the converse is also completely false.
So something has gone wrong here. Explain what.
Class Problems
Problem 3.29.
A media tycoon has an idea for an all-news television network called LNN: The
Logic News Network. Each segment will begin with a definition of the domain of
discourse and a few predicates. The day’s happenings can then be communicated
concisely in logic notation. For example, a broadcast might begin as follows:
(a)
(b)
THIS IS LNN. The domain of discourse is
fAlbert; Ben; Claire; David; Emilyg:
Let D.x/ be a predicate that is true if x is deceitful. Let L.x; y/
be a predicate that is true if x likes y. Let G.x; y/ be a predicate that
is true if x gave gifts to y.
Translate the following broadcasts in logic notation into (English) statements.
NOT.D.Ben/ OR D.David// IMPLIES
.L.Albert; Ben/ AND L.Ben; Albert//:
8x: ..x D Claire AND NOT.L.x; Emily/// OR .x ¤ Claire AND L.x; Emily///
AND
8x: ..x D David AND L.x; Claire// OR .x ¤ David AND NOT.L.x; Claire////