Lecture 4: Predicates and Quantifiers; Sets.. L42 Agenda Predicates and Quantifiers Existential...
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Transcript of Lecture 4: Predicates and Quantifiers; Sets.. L42 Agenda Predicates and Quantifiers Existential...
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AgendaPredicates and Quantifiers Existential Quantifier Universal Quantifier
Sets 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 exampleConsider the compound proposition:
If the Lion is an octopus then the Lion has 8 limbs.
Q1: What are the atomic propositions.Q2: Is the proposition true or false?Q3: Why?
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Motivating exampleA1: Let p = “The Lion is an octopus” and
q = “the Lion has 8 limbs”. The compound proposition is represented by p q.
A2: True!A3: Conditional always true when p is false!
Q: How can we represent: it exists an animal such that it has 8 limbs?
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Motivating exampleLogical 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 exampleThe 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 In order to add to propositions with
meaning, we need to have a universe of discourse, i.e. a collection of subjects (or nouns) about which the propositions relate.
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PredicatesIf we have a statement involving some
variables, such asx>3 or x=y+2 or x+y=zThese statements are consist of some
variables x,y and z and a predicate (such as greater than or equal) which gives the statement some truth value in the universe of discourse where x,y and z exist .
Predicates…… continue
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. 7 is a prime number
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Propositional FunctionsBy 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 n is a prime number. …e.g. plug in n = 111
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Quantifiers
There are two quantifiersExistential 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
“x P (x)” is true when an instance can be found which when plugged in for x makes P (x) true
Like disjunction 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
“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 TF gives F, and the conjunction is false.
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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 ExampleA: 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 and that y - x > z ?
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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 mattersR (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 + 1y 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 RevisitedRecall DeMorgan’s identities:
Conjunctional negation:(p1p2…pn) (p1p2…pn)
Disjunctional negation:(p1p2…pn) (p1p2…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 yIn 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 ExercisesSection 1.3
1.3.42) Show that the following are logically equivalent:(x A(x ))(x B(x ))
x,y A(x )B(y )
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Blackboard ExercisesSection 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 ) trueCase 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.Thusx 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.
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SetsCurly braces “{“ and “}” are used to denote
sets.
EG: { 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: { 11, 11, 11, 12, 13 } = { 11, 12, 13 } { , , } = { , , }
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Standard Numerical SetsThe 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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-NotationThe 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:1. 3 R2. -3 N3. -3 R4. 0 Z+
5. x xR x2=-5
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-Notation
A: 1, 3 and 41. 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 xR x2=-5 . False: square of a real number is non-neg., so can’t be -5.
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-NotationDEF: 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 TA 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 (xS ) (xT )We already had all the necessary concepts, but
the “” notation saves work.
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-NotationWhen “” 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 xT )
Q: What algebraic symbol is reminiscent of?
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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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Subset Examples
A: 1, 4 and 51. N R. All natural numbers are real.2. Z N. Negative numbers aren’t natural.3. -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).
4. {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.
5. . Any set contains itself.6. . No set can contain itself properly.
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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. |{1, -13, 4, -13, 1}|2. |{3, {1,2,3,4}, }|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. |{1, -13, 4, -13, 1}| = |{1, -13, 4}| = 32. |{3, {1,2,3,4}, }| = 3. To see this, set S =
{1,2,3,4}. Compute the cardinality of {3,S, }
3. |{}| = || = 04. |{ {}, {{}}, {{{}}} }|
= |{ , {}, {{}}| = 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 –ExampleTo 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} +32-element sets: {1,2}, {1,3}, {2,3} +33-element sets: {1,2,3} +1Therefore: | 2s | = 8 = 23 = 2|s|
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Ordered n-tuplesNotationally, 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
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Ordered n-tuples
As opposed to sets, repetition and ordering do matter with n-tuples. (11, 11, 11, 12, 13) ( 11, 12, 13 ) ( , , ) ( , , )
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Cartesian ProductThe 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 x-axis 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 aA and bB .
Q: Describe R2 as the Cartesian product of two sets.
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Cartesian ProductA: R2 = RR. 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 ProductA: 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?