70 lines
1.7 KiB
Text
70 lines
1.7 KiB
Text
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#import "./dvd.typ": *
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#show: dvdtyp.with(
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title: "Math 8",
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subtitle: [UC Santa Barbara],
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author: "Youwen Wu",
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)
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#outline()
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= Chapter 1: Logic and Proofs
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== Trivial Preliminaries
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Definitions barely worth considering. Included purely for posterity.
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#definition("Proposition")[
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A proposition is a sentence which is either true or false.
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]
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#example("Primes")[
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The numbers 5 and 7 are prime.
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]
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#example("Not a proposition")[
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$x^2 + 6x + 8 = 0$
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]
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Propositions may be stated in the formalism of mathematics using connectives, as *propositional forms*.
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#definition("Propositional forms")[
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Let $P$ and $Q$ be propositions. Then:
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+ The conjunction of $P$ and $Q$ is written $P and Q$ ($P$ and $Q$).
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+ The disjunction of $P$ and $Q$ is written $P or Q$ ($P$ or $Q$) (here "or" is the inclusive or).
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+ The negation of $P$ is written $not P$.
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]
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#definition("Tautology")[
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A propositional form for which all of its values are true. In other words, a statement which is always true.
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]
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#definition("Contradiction")[
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A propositional form for which all of its values are false. In other words, a statement which is always false.
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]
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#problem[Prove that $(P or Q) or (not P and not Q)$ is a tautology][
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Trivial, omitted.
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]
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#example[Several denials of the statement "integer $n$ is even"][
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- It is not the case that integer $n$ is even.
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- Integer $n$ is not even.
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- $n != 2m, forall m in ZZ$
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- $n = 2m + 1, exists m in ZZ$
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]
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DeMorgan's Laws tell us how to distribute logical connectives across parentheses.
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#theorem[DeMorgan's Laws][
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+ $not (P or Q) = not P and not Q$
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+ $not (P and Q) = not P or not Q$
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]
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#proof[
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Trivially, by completing a truth table.
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]
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