SYMBOLIC LOGIC
(PART TWO)
LAMC INTERMEDIATE - 6/1/14
WARM U P
(1) Suppose we have a triangle ABC. Let MA be the midpoint of side BC, MB
be the midpoint of side AC, and MC be the midpoint of side AB. Express the
!
!
following in terms of the vectors ~v = AB and w
~ = AC. (DRAW A PICTURE!!)
!
(a) AMA
!
(b) BMB
!
(c) CMC
Copyright c 2008-2014 Olga Radko/Los Angeles Math Circle/UCLA Department of Mathematics .
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(2) The notation from the previous problem continues here. Compute each of the
following in terms of ~v and w:
~
!
(a) 23 AMA
!
(b) ~v + 23 BMB
!
(c) w
~ + 23 CMC
(3) Prove that all three medians of a triangle intersect at a single point.
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(4) (Midpoint theorem) Let ABC be a triangle. Let M be the midpoint of AB and
N be the midpoint of AC. Show that the length of M N is half the length
of BC. (Hint: Express each side of the triangle with vectors and show that
!
!
M N = 12 BC!)
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N EGATION
AND
D E M ORGAN ’ S L AWS
(1) Show that ¬ (P _ Q) is logically equivalent to ¬P ^ ¬Q.
(2) Show that ¬ (P ^ Q) is logically equivalent to ¬P _ ¬Q.
(3) Prove or disprove: ¬(P ! Q) is logically equivalent to ¬P ! ¬Q.
(4) Prove or disprove: ¬(P ! Q) is logically equivalent to P ^ ¬Q.
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S ATISFIABILITY
A formula (or collection of formulas) is called satisfiable if there is a model in which
the formula (or collection of formulas) is true. For us, a model is simply a truth
assignment to all propositional variables. For example, if P, Q, and R are our propositional variables a model could be
P Q R
T F T
In this model P ^ Q is false, but P ^ R is true. A formula is thus satisfiable if we can
come up with a model in which it is true.
(1) Let P, Q, and R be propositional variables. Determine whether or not the following are satisfiable. If you think a formula is satisfiable, provide a model
which satisfies it.
(a) ¬Q _ Q
(b) ¬P ^ P
(c) ¬ (P ! Q) ^ R
(d) (P ! Q) $ (¬Q ! P )
(e) (P ^ ¬P ) _ ¬Q
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(f) {¬ (P ! Q) ^ ¬R, (P ^ ¬R) _ ¬Q}
(g) {¬ (P ! Q) , ¬R, (P _ Q) $ R}
(h) {(P ^ ¬R) _ ¬Q, (P _ Q) $ R}
(i) {P ! Q, ¬Q ! ¬P, ¬P _ Q}
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The Compactness Theorem. A set of formulas has a model if and only if every
finite subset has a model. (For the remainder of the handout, we will assume this
theorem is true)
(1) Prove the forward direction of the Compactness Theorem. That is, prove that
if a set of formulas has a model, then every finite subset has a model.
(2) (Non-Standard Analysis) In this problem, we will construct a model of the
real numbers which contains infinitesimally small numbers. (Such numbers
are the basis for how Leibniz perceived of calculus)
(a) Let ⌃ be a set of formulas which defines the real numbers. Let ✏ be a new
symbol. Let n be arbitrary and consider the proposition ✏ < n1 . Is this
proposition satisfiable?
(b) Show that the propositions ✏ < 1, · · · , ✏ <
1
n
are satisfiable.
(c) Use the compactness theorem to conclude that the collection ✏ <
is satisfiable.
1
n
:n2N
(d) Prove that there is a model of the real numbers with infinitely large elements.
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T AUTOLOGIES
AND
T AUTOLOGICAL I MPLICATION
We say that a set of formulas, ⌃, tautologically implies a formula ✓, written ⌃ |= ✓,
if every model of ⌃ is also a model of ✓. We say that a formula ✓ is a tautology if
every set of formulas tautologically implies ✓.
(1) Show that
{P, P ! ¬Q, Q $ R} |= ¬R.
(2) Show that
{P, ¬P } |= ((Q ! P ) ^ (R _ (Q $ ¬P ))) ! ¬(Q $ R)
(3) Show that
{P ^ (Q ! R) , ¬R} |= ¬Q
(4) Come up with a tautology.
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(5) Of the following formulas, which imply which? (That is, does {✓1 } |= ✓2 ? etc. )
(a) ✓1 = (P ! Q)
(b) ✓2 = ¬ ((P ! Q) ! ¬ (Q ! P ))
(c) ✓3 = (¬P _ Q) ^ (P _ ¬Q)
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(6) Prove the following:
(a) (Deduction Theorem) ⌃ [ {✓} |= ' if and only if ⌃ |= ✓ ! '.
(b) If ⌃ |= ' or ⌃ |= ✓, then ⌃ |= ' _ ✓.
(c) Is the converse of (b) true? That is, if ⌃ |= ' _ ✓ does ⌃ |= ' or ⌃ |= ✓? Prove
or provide a counterexample.
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C HALLENGE P ROBLEMS
(1) Suppose that ⌃ is a set of propositions. Suppose further that for every model
M there is some formula 2 ⌃ such that M |= . Prove that there are formulas
1 , · · · , n 2 ⌃ such that
1
is a tautology.
_
2
_ ··· _
n
(2) Prove the following theorem: If ⌃ |= p then there is a finite subset ⌃0 of ⌃ such
that ⌃0 |= p.
Math Circle (Intermediate) 6/1/2014
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