On Oct 26, 2010, at 12:43 PM, Andrew Coppin wrote:
Propositional logic is quite a simple logic, where the building
blocks
are atomic formulae and the usual logical connectives. An example
of a
well-formed formula might be "P → Q". It tends to be the first
system
taught to undergraduates, while the second is usually the first-order
predicate calculus, which introduces predicates and quantifiers.
Already I'm feeling slightly lost. (What does the arrow denote?
What's are "the usual logcal connectives"?)
The arrow is notation for "If P, then Q". The other "usual" logical
connectives are "not" (denoted by ~, !, the funky little sideways L,
and probably others); "or" (denoted by \/, v, (both are pronounced
"or" or "vee" even "meet") |, ||, and probably others);
"and" (denoted by /\, or a smaller upside-down v (pronounced "wedge"
or "and" or even "join"), &, &&, and probably others).
Predicates are usually interpreted as properties; we might write
"P(x)" or "Px" to indicate that object x has the property P.
Right. So a proposition is a statement which may or may not be true,
while a predicate is some property that an object may or may not
possess?
Yes. For any given object a (which is not a "variable" -- we usually
reserve x, y, z to denote variables, and objects are denoted by a, b,
c), P(a) is a proposition "about" a. Something like "forall x P(x)"
means that P(x) is true for every object in the domain you are
considering.
I also don't know exactly what "discrete mathematics" actually
covers.
Discrete mathematics is concerned with mathematical structures which
are discrete, rather than continuous.
Right... so its domain is simply *everything* that is discrete? From
graph theory to cellular automina to finite fields to difference
equations to number theory? That would seem to cover approximately
50% of all of mathematics. (The other 50% being the continuous
mathematics, presumably...)
Basically, yes. There are some nuances, in that continuous structures
might be studied in terms of discrete structures, and vice-versa. _______________________________________________
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