Transcript Lecture 4: Predicates and Quantifiers; Sets. Zeph Grunschlag Copyright © Zeph Grunschlag, 2001-2002. Announcements HW1 due now L4

Lecture 4: Predicates and
Quantifiers; Sets.
Zeph Grunschlag
Copyright © Zeph Grunschlag,
2001-2002.
Announcements
HW1 due now
L4
2
Agenda
Predicates and Quantifiers


Existential Quantifier 
Universal Quantifier 
Sets







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Curly brace notation “{ … }”
Cardinality “| … |”
Element containment 
Subset containment 
Empty set { } = 
Power set P(S ) = 2S
N-tuples “( … )” and Cartesian product 
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Motivating example
Consider the compound proposition:
If Zeph is an octopus then Zeph has 8 limbs.
Q1: What are the atomic propositions and how
do they form this proposition.
Q2: Is the proposition true or false?
Q3: Why?
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Motivating example
A1: Let p = “Zeph is an octopus” and q =
“Zeph has 8 limbs”. The compound
proposition is represented by p q.
A2: True!
A3: Conditional always true when p is false!
Q: Why is this not satisfying?
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Motivating example
A: We wanted this to be true because of the fact that
octopi have 8 limbs and not because of some
(important) non-semantic technicality in the truth
table of implication.
But recall that propositional calculus doesn’t take
semantics into account so there is no way that p
could impact on q or affect the truth of pq.
Logical Quantifiers help to fix this problem. In our case
the fix would look like:
For all x, if x is an octopus then x has 8 limbs.
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Motivating example
Expressions such as the previous are built up from
propositional functions –statements that have a
variable or variables that represent various possible
subjects. Then quantifiers are used to bind the
variables and create a proposition with embedded
semantics. For example:
“For all x, if x is an octopus then x has 8 limbs.”
there are two atomic propositional functions


P (x) = “x is an octopus”
Q (x) = “x has 8 limbs”
whose conditional P (x) Q (x) is formed and is
bound by “For all x ”.
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Semantics
If logical propositions are to have meaning,
they need to describe something. Up to now,
propositions such as “Johnny is tall”, “Debbie
is 5 years old”, and “Andre is immature” had
no intrinsic meaning. Either they are true or
false, but no more.
In order to endow such propositions with
meaning, we need to have a universe of
discourse, i.e. a collection of subjects (or
nouns) about which the propositions relate.
Q: What is the universe of discourse for the
three propositions above?
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Semantics
A: There are many answers. Here are some:



Johnny, Debbie and Andre (this is also the
smallest correct answer)
People in the world
Animals
Java: The Java analog of a universe of
discourse is a type. There are two
categories of types in Java: reference types
which consist of objects and arrays, and
primitive types like int, boolean, char,
etc. Examples of Java “universes” are:
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int, char, int[][], Object, String,
java.util.LinkedList, Exception, etc.
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Predicates
A predicate is a property or description of
subjects in the universe of discourse. The
following predicates are all italicized :



Johnny is tall.
The bridge is structurally sound.
17 is a prime number.
Java: predicates are boolean-valued method
calls

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someLinkedList.isEmpty()
isPrime(17)
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Propositional Functions
By taking a variable subject denoted by symbols
such as x, y, z, and applying a predicate one
obtains a propositional function (or
formula). When an object from the universe
is plugged in for x, y, etc. a truth value
results:



x is tall. …e.g. plug in x = Johnny
y is structurally sound. …e.g. plug in y = GWB
n is a prime number. …e.g. plug in n = 111
Java: propositional functions are boolean
methods, rather than particular calls.

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
boolean isEmpty(){…} //in LinkedList
boolean isPrime(int n){…}
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Multivariable Predicates
Multivariable predicates generalize
predicates to allow descriptions of
relationships between subjects. These
subjects may or may not even be in the same
universe of discourse. For example:



Johnny is taller than Debbie.
17 is greater than one of 12, 45.
Johnny is at least 5 inches taller than Debbie.
Q: What universes of discourse are involved?
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Multivariable Predicates
A: Again, many correct answers. The most
obvious answers are:



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For “Johnny is taller than Debbie” the universe of
discourse of both variables is all people in the
universe
For “17 is greater than one of 12, 45” the universe
of discourse of all three variables is Z (the set of
integers)
For “Johnny is at least 5 inches taller than Debbie”
the first and last variable have people as their
universe of discourse, while the second variable
has R (the set of real numbers).
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Multivariable Propositional
Functions
The multivariable predicates, together with their
variables create multivariable propositional
functions. In the above examples, we have
the following generalizations:



x is taller than y
a is greater than one of b, c
x is at least n inches taller than y
In Java, a multivariable predicate is a boolean
method with several arguments:
tallerByNumInches(Person x, double n, Person y)
{ … }
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Quantifiers
There are two quantifiers
Existential Quantifier
“” reads “there exists”
Universal Quantifier
“” reads “for all”
Each is placed in front of a propositional
function and binds it to obtain a
proposition with semantic value.
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Existential Quantifier


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“x P (x)” is true when an instance can be
found which when plugged in for x makes
P (x) true
Like disjunctioning over entire universe
x P (x )  P (x1) P (x2) P (x3)  …
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Existential Quantifier.
Example
Consider a universe consisting of
 Leo: a lion
 Jan: an octopus with all 8 tentacles
 Bill: an octopus with only 7 tentacles
And recall the propositional functions
 P (x) = “x is an octopus”
 Q (x) = “x has 8 limbs”
x ( P (x) Q (x) )
Q: Is the proposition true or false?
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Existential Quantifier.
Example
A: True. Proposition is equivalent to
(P (Leo)Q (Leo) )(P (Jan) Q (Jan) )(P(Bill)Q (Bill) )
P (Leo) is false because Leo is a Lion, not
an octopus, therefore the conditional
P (Leo) Q (Leo) is true, and the
disjunction is true.
Leo is called a positive example.
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The Universal Quantifier


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“x P (x)” true when every instance of x
makes P (x) true when plugged in
Like conjunctioning over entire universe
x P (x )  P (x1) P (x2)  P (x3)  …
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Universal Quantifier. Example
Consider the same universe and
propositional functions as before.
 x ( P (x) Q (x ) )
Q: Is the proposition true or false?
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Universal Quantifier. Example
A: False. The proposition is equivalent to
(P (Leo)Q (Leo))(P (Jan)Q (Jan))(P (Bill)Q (Bill))
Bill is the counter-example, i.e. a value making
an instance –and therefore the whole universal
quantification– false.
P (Bill) is true because Bill is an octopus, while
Q (Bill) is false because Bill only has 7 tentacles,
not 8. Thus the conditional P (Bill)Q (Bill) is
false since TF gives F, and the conjunction is
false.
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Illegal Quantifications
Once a variable has been bound, we cannot
bind it again. For example the expression
x ( x P (x) )
is nonsensical. The interior expression
(x P (x)) bounded x already and therefore
made it unobservable to the outside. Going
back to our example, the English equivalent
would be:
Everybody is an everybody is an octopus.
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Multivariate Quantification
Quantification involving only one variable
is fairly straightforward. Just a bunch
of OR’s or a bunch of AND’s.
When two or more variables are involved
each of which is bound by a quantifier,
the order of the binding is important
and the meaning often requires some
thought.
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Parsing Example
A: True.
For any “exists” we need to find a positive
instance.
Since x is the first variable in the expression
and is “existential”, we need a number that
works for all other y, z. Set x = 0 (want to
ensure that y -x is not too small).
Now for each y we need to find a positive
instance z such that y - x ≥ z holds.
Plugging in x = 0 we need to satisfy y ≥ z so
set z := y.
Q: Did we have to set z := y ?
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Parsing Example
A: No. Could also have used the
constant z := 0. Many other valid
solutions.
Q: Isn’t it simpler to satisfy
x y z (y - x ≥ z )
by setting x := y and z := 0 ?
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Parsing Example
A: No, this is illegal ! The existence of x
comes before we know about y. I.e.,
the scope of x is higher than the scope
of y so as far as y can tell, x is a
constant and cannot affect x.
A Java example helps explain this point.
To evaluate x y Q (x,y ) might do the
following.
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Parsing Example —Java
boolean exists_x_forAll_y(){
for(int x=firstInt; x<=lastInt; x++){
if (forAll_y(x)) return true;
}
return false;
}
boolean forAll_y(int x){
for(int y=firstInt; y<=lastInt; y++){
if ( !Q(x,y) ) return false;
}
return true;
}
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Order matters
Set the universe of discourse to be all
natural numbers {0, 1, 2, 3, … }.
Let R (x,y ) = “x < y”.
Q1: What does x y R (x,y ) mean?
Q2: What does y x R (x,y ) mean?
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Order matters
R (x,y ) = “x < y”
A1: x y R (x,y ):
“All numbers x admit a bigger number y ”
A2: y x R (x,y ):
“Some number y is bigger than all x”
Q: What’s the true value of each
expression?
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Order matters
A: 1 is true and 2 is false.
x y R (x,y ): “All numbers x admit a bigger
number y ” --just set y = x + 1
y x R (x,y ): “Some number y is bigger than
all numbers x” --y is never bigger than itself,
so setting x = y is a counterexample
Q: What if we have two quantifiers of the same
kind? Does order still matter?
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Order matters –but not always
A: No! If we have two quantifiers of the
same kind order is irrelevent.
x y is the same as y x because
these are both interpreted as “for every
combination of x and y…”
x y is the same as y x because
these are both interpreted as “there is a
pair x , y…”
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Logical Equivalence with
Formulas
DEF: Two logical expressions possibly involving
propositional formulas and quantifiers are
said to be logically equivalent if no-matter
what universe and what particular
propositional formulas are plugged in, the
expressions always have the same truth
value.
EG: x y Q (x,y ) and y x Q (y,x ) are
equivalent –names of variables don’t matter.
EG: x y Q (x,y ) and y x Q (x,y ) are not!
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DeMorgan Revisited
Recall DeMorgan’s identities:
Conjunctional negation:
(p1p2…pn)  (p1p2…pn)
Disjunctional negation:
(p1p2…pn)  (p1p2…pn)
Since the quantifiers are the same as taking a
bunch of AND’s () or OR’s () we have:
Universal negation:
 x P(x )  x P(x )
Existential negation:
 x P(x )  x P(x )
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Negation Example
Compute:
 x y x2  y
In English, we are trying to find the opposite
of “every x admits a y greater or equal to
x’s square”. The opposite is that “some x
does not admit a y greater or equal to x’s
square”
Algebraically, one just flips all quantifiers
from  to  and vice versa, and negates
the interior propositional function. In our
case we get:
x y ( x 2  y )  x y x 2 > y
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Blackboard Exercises
Section 1.3
1.3.41) Show that the following are
logically equivalent:
(x A(x ))(x B(x ))
x,y A(x )B(y )
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Blackboard Exercises
Section 1.3
Need to show that
(x A(x ))(x B(x ))x,y A(x )B(y )
is a tautology, so LHS and RHS must always
have same truth values.
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Blackboard Exercises Section 1.3
(x A(x ))(x B(x ))x,y A(x )B(y )
CASE I) Assuming LHS true show RHS true.
Either x A(x ) true OR x B(x ) true
Case I.A) For all x, A(x ) true.
As (T  anything) = T, we can set
anything = B(y) and obtain
For all x and y, A(x )B(y ) –the RHS!
Case I.B) For all x, B(x ) true… similar to
case (I.A)
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Blackboard Exercises Section 1.3
(x A(x ))(x B(x ))x,y A(x )B(y )
CASE II) Assume LHS false, show RHS false.
Both x A(x ) false AND x B(x ) false.
Thus x  A(x ) true AND x  B(x ) true.
Thus there is an example x1 for which A(x1) is false
and an example x2 for which B(x2) is false.
As F  F = F, we have A(x1)B(x2) false.
Setting x = x1 and y = x2 gives a counterexample
to x,y A(x )B(y ) showing the RHS false.
//QED
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Section 1.4: Sets
DEF: A set is a collection of elements.
This is another example where mathematics
must start at the level of intuition. Sets are
the basic data structure out of which most
mathematical theories are built. For many
years mathematicians hoped that sets could
be defined directly from logic, thus giving a
full-proof foundation to Mathematics, when
compared to other sciences. Effort failed!
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Sets
Curly braces “{“ and “}” are used to denote
sets.
Java note: In Java curly braces denote arrays,
a data-structure with inherent ordering.
Mathematical sets are unordered so different
from Java arrays. Java arrays require that all
elements be of the same type. Mathematical
sets don’t require this, however. EG:



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{ 11, 12, 13 }
{
,
,
{
,
,
}
, 11, Leo }
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Sets
A set is defined only by the elements
which it contains. Thus repeating an
element, or changing the ordering of
elements in the description of the set,
does not change the set itself:


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{ 11, 11, 11, 12, 13 } = { 11, 12, 13 }
{
,
,
}={
,
,
}
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Standard Numerical Sets
The natural numbers:
N = { 0, 1, 2, 3, 4, … }
The integers:
Z = { … -3, -2, -1, 0, 1, 2, 3, … }
The positive integers:
Z+ = {1, 2, 3, 4, 5, … }
The real numbers: R --contains any decimal
number of arbitrary precision
The rational numbers: Q --these are decimal
numbers whose decimal expansion repeats
Q: Give examples of numbers in R but not Q.
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Standard Numerical Sets
A:
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2 , π, e, or any irrational number
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-Notation
The Greek letter “” (epsilon) is used to denote
that an object is an element of a set. When
crossed out “” denotes that the object is
not an element.”
EG: 3  S reads:
“3 is an element of the set S ”.
Q: Which of the following are true:
4.
3R
-3  N
-3  R
0  Z+
5.
x xR
1.
2.
3.
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
x2=-5
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-Notation
A: 1, 3 and 4
1. 3  R. True: 3 is a real number.
2. -3  N. False: natural numbers don’t
contain negatives.
3. -3  R. True: -3 is a real number.
4. 0  Z+. True: 0 isn’t positive.
5. x xR  x2=-5 . False: square of a
real number is non-neg., so can’t be -5.
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-Notation
DEF: A set S is said to be a subset of the set T
iff every element of S is also an element of
T. This situation is denoted by
ST
A synonym of “subset” is “contained by”.
Definitions are often just a means of
establishing a logical equivalence which aids
in notation. The definition above says that:
ST

x (xS )  (xT )
We already had all the necessary concepts, but
the “” notation saves work.
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-Notation
When “” is used instead of “”, proper
containment is meant. A subset S of T is
said to be a proper subset if S is not equal
to T. Notationally:
ST
 S T
 x
(x  S
 xT )
Q: What algebraic symbol is  reminiscent of?
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-Notation
A:  is to , as < is to .
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The Empty Set
The empty set is the set containing no
elements. This set is also called the
null set and is denoted by:


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{}

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Subset Examples
Q: Which of the following are true:
1.
2.
3.
4.
5.
6.
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NR
ZN
-3  R
{1,2}  Z+


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Subset Examples
A: 1, 4 and 5
1.
2.
3.
4.
5.
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6.
N  R. All natural numbers are real.
Z  N. Negative numbers aren’t natural.
-3  R. Nonsensical. -3 is not a subset but an
element! (This could have made sense if we
viewed -3 as a set –which in principle is the
case– in this case the proposition is false).
{1,2}  Z+. This actually makes sense. The set
{1,2} is an object in its own right, so could be
an element of some set; however, {1,2} is not a
number, therefore is not an element of Z.
  . Any set contains itself.
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  . No set can contain itself properly.
Cardinality
The cardinality of a set is the number of
distinct elements in the set. |S |
denotes the cardinality of S.
Q: Compute each cardinality.
1.
2.
3.
4.
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|{1, -13, 4, -13, 1}|
|{3, {1,2,3,4}, }|
|{}|
|{ {}, {{}}, {{{}}} }|
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Cardinality
Hint: After eliminating the redundancies just
look at the number of top level commas and
add 1 (except for the empty set).
A:
1.
2.
3.
4.
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|{1, -13, 4, -13, 1}| = |{1, -13, 4}| = 3
|{3, {1,2,3,4}, }| = 3. To see this, set S =
{1,2,3,4}. Compute the cardinality of {3,S, }
|{}| = || = 0
|{ {}, {{}}, {{{}}} }|
= |{ , {}, {{}}| = 3
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Cardinality
DEF: The set S is said to be finite if its
cardinality is a nonnegative integer.
Otherwise, S is said to be infinite.
EG: N, Z, Z+, R, Q are each infinite.
Note: We’ll see later that not all infinities are
the same. In fact, R will end up having a
bigger infinity-type than N, but surprisingly,
N has same infinity-type as Z, Z+, and Q.
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Power Set
DEF: The power set of S is the set of all
subsets of S.
Denote the power set by P (S ) or by 2s .
The latter weird notation comes from the
following.
Lemma: | 2s | = 2|s|
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Power Set –Example
To understand the previous fact consider
S = {1,2,3}
Enumerate all the subsets of S :
0-element sets: {}
1
1-element sets: {1}, {2}, {3}
+3
2-element sets: {1,2}, {1,3}, {2,3}
+3
3-element sets: {1,2,3}
+1
Therefore: | 2s | = 8 = 23 = 2|s|
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Ordered n-tuples
Notationally, n-tuples look like sets except that
curly braces are replaced by parentheses:



( 11, 12 ) –a 2-tuple aka ordered pair
(
,
,
) –a 3-tuple
(
,
,
, 11, Leo ) –a 5-tuple
Java: n -tuples are similar to Java arrays “{…}”,
except that type-mixing isn’t allowed in Java.
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Ordered n-tuples
As opposed to sets, repetition and
ordering do matter with n-tuples.


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(11, 11, 11, 12, 13)  ( 11, 12, 13 )
(
,
,
)(
,
,
)
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Cartesian Product
The most famous example of 2-tuples are
points in the Cartesian plane R2. Here
ordered pairs (x,y) of elements of R describe
the coordinates of each point. We can think
of the first coordinate as the value on the xaxis and the second coordinate as the value
on the y-axis.
DEF: The Cartesian product of two sets A
and B –denoted by A B– is the set of all
ordered pairs (a, b) where aA and bB .
Q: Describe R2 as the Cartesian product of two
sets.
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Cartesian Product
A: R2 = RR. I.e., the Cartesian plane is
formed by taking the Cartesian product
of the x-axis with the y-axis.
One can generalize the Cartesian product
to several sets simultaneously.
Q: If A = {1,2}, B = {3,4}, C = {5,6,7}
what is A B C ?
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Cartesian Product
A: A = {1,2}, B = {3,4}, C = {5,6,7}
A B C =
{ (1,3,5), (1,3,6), (1,3,7),
(1,4,5), (1,4,6), (1,4,7),
(2,3,5), (2,3,6), (2,3,7),
(2,4,5), (2,4,6), (2,4,7) }
Lemma: The cardinality of the Cartesian
product is the product of the cardinalities:
| A1  A2  … An | = |A1||A2| … |An|
Q: What does S equal?
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Cartesian Product
A: From the lemma:
|S | = |||S | = 0|S | = 0
There is only one set with no elements –
the empty set– therefore, S must be
the empty set .
One can also check this directly from the
definition of the Cartesian product.
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Blackboard Exercise
Section 1.4
Prove the following:
If A  B and B  C then A  C .
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