Week 1 - Friday
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What did we talk about last time?
Implications
 Inverses
 Converses
 Bidirectional
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Tautologies
Contradictions
Laws of Boolean algebra
Rules of inference
Digital logic circuits
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For Diwali, Mr. Patel's five daughters gave
each other books as presents.
Each presented four books and each received
four books, but no two girls divided her books
in the same way.
That is, only one gave two books to one sister
and two to another. Bharat gave all her
books to Abhilasha; Chandra gave three to
Esha.
Who gave how many books to whom?
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One note about implications and wording them:
 p is a sufficient condition for q means p  q
 p is a necessary condition for q means q  p
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This nomenclature is a touch counterintuitive
Think of it this way:
 p  q means that p is enough to get you q, but there
might be other things that will get you q
 q  p means that, since you automatically get p when
you've got q, there's no way to have q without p
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An argument is a list of statements (called
premises) followed by a single statement
(called a conclusion)
Whenever all of the premises are true, the
conclusion must also be true, in order to
make the argument valid
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Are the following arguments valid?
 p  q  ~r
 qpr
 pq
 p  (q  r)
 ~r
 pq
(premise)
(premise)
(conclusion)
(premise)
(premise)
(conclusion)
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Modus ponens is a valid argument of the following
form:
 pq
 p
 q
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Modus tollens is a contrapositive reworking of the
argument, which is also valid:
 pq
 ~q
  ~p
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Give verbal examples of each
We call these short valid arguments rules of inference
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The following are also valid rules of inference:
p
  p q
q
  p q
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English example: “If pigs can fly, then pigs
can fly or swans can breakdance.”
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The following are also valid rules of inference:
 pq
 p
 pq
 q
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English example: “If the beat is out of control
and the bassline just won’t stop, then the
beat is out of control.”
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The following is also a valid rule of inference:
p
q
 pq
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English example: “If the beat is out of control
and the bassline just won’t stop, then the
beat is out of control and the bassline just
won’t stop.”
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The following are also valid rules of inference:
 pq
 ~q
 p
 pq
 ~p
 q
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English example: “If you’re playing it cool or I’m
maxing and relaxing, and you’re not playing it cool,
then I’m maxing and relaxing.”
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The following is also a valid rule of inference:
 pq
 qr
 pr
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English example: “If you call my mom ugly I
will call my brother, and if I call my brother he
will beat you up, then if you call my mom ugly
my brother will beat you up.”
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The following is also a valid rule of inference:
 pq
 pr
 qr
 r
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English example: “If am fat or sassy, and
being fat implies that I will give you trouble,
and being sassy implies that I will give you
trouble, then I will give you trouble.”
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The following is also a valid rule of inference:
 ~p  c
 p
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English example: “If my water is at absolute
zero then the universe does not exist, thus
my water must not be at absolute zero.”
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First, convert the following statements into symbolic
logic
If this house is next to a lake, then the treasure is not in
the kitchen.
b) If the tree in the front yard is an elm, then the treasure is
in the kitchen.
c) This house is next to a lake.
d) The tree in the front yard is an elm or the treasure is
buried under the flagpole.
e) If the tree in the back yard is an oak, then the treasure is
in the garage.
a)
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Then, apply appropriate rules of inference to
determine where the treasure is hidden
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First, convert the following statements about the murder of Lord
Hazelton into symbolic logic
a)
b)
c)
d)
e)
f)
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Lord Hazelton was killed by a blow on the head with a brass
candlestick.
Either Lady Hazelton or a maid, Sara, was in the dining room at the
time of the murder.
If the cook was in the kitchen at the time of the murder, then the
butler killed Lord Hazelton with a fatal dose of strychnine
If Lady Hazelton was in the dining room at the time of the murder,
then the chauffeur killed Lord Hazelton.
If the cook was not in the kitchen at the time of the murder, then
Sara was not in the dining room when the murder was committed.
If Sara was in the dining room at the time the murder was
committed, then the wine steward killed Lord Hazelton.
Then, apply appropriate rules of inference to determine who the
murderer is (if possible), assuming only one cause of death
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Draw the digital logic circuit corresponding
to:
p  (q  ~r)
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A predicate is a sentence with a fixed number
of variables that becomes a statement when
specific values are substituted for to the
variables
The domain gives all the possible values that
can be substituted
The truth set of a predicate P(x) are those
elements of the domain that make P(x) true
when they are substituted
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Let P(x) be “x has had 4 wisdom teeth
removed”
What is the truth set if the domain is the
people in this classroom?
Let Q(n) be “n is divisible by exactly itself and
1”
What is the truth set if the domain is the set
of positive integers Z+?
We will frequently be referring to various sets of numbers
in this class
 Some typical notation used for these sets:
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Symbol
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Set
Examples
R
Real numbers
Virtually everything that isn’t imaginary
Z
Integers
{…, -2, -1, 0, 1, 2,…}
Z-
Negative integers
{-1, -2, -3, …}
Z+
Positive integers
{1, 2, 3, …}
N
Natural numbers
{1, 2, 3, …}
Q
Rational numbers
a/b where a,b  Z and b  0
Some authors use Z+ to refer to non-negative integers
and only N for the natural numbers
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The universal quantifier  means “for all”
The statement “All DJ’s are mad ill” can be
written more formally as:
x  D, M(x)
 Where D is the set of DJ’s and M(x) denotes that x
is mad ill
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Let S = {1, 2, 3, 4, 5}
Show that the following statement is true:
 x  S, x2 ≥ x
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Show that the following statement is false:
 x  R, x2 ≥ x
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The universal quantifier  means “there
exists”
The statement “Some emcee can bust a
rhyme” can be written more formally as:
y  E, B(y)
 Where E is the set of emcees and B(y) denotes
that y can bust a rhyme
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Let S = {2, 4, 6, 8}
Show that the following statement is false:
 x  S, 1/x = x
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Show that the following statement is true:
  x  Z, 1/x = x
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Tarski’s World provides an easy framework for
testing knowledge of quantifiers
The following notation is used:
 Triangle(x) means “x is a triangle”
 Blue(y) means “y is blue”
 RightOf(x, y) means “x is to the right of y (but not
necessarily on the same row)”
a
b
c
e
f
g
h
d
i
j
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k
Are the following statements true or false?
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t, Triangle(t)  Blue(t)
x, Blue(x)  Triangle(x)
y such that Square(y)  RightOf(d, y)
z such that Square(z)  Gray(z)
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Negating quantifications
Multiple quantifications
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Keep reading Chapter 3
Start working on Assignment 1
 Due next Friday