MATH135 week 2 Sets (ctd), Quantifiers, Nested Quantifiers, Induction Sept. 16-20, 2013
Quantifiers
In math, we have a couple of key words/phrases used in small statements. Depending on the
quantifier used certain proof techniques can be used to prove the statement.
Existential quantifiers: There are, there is, there exists
These are existential quantifiers and we denote them wWe used ∃ when we
are looking for
math objects.
Basic Structure: There is some ‘object’ in the ‘set where the object was taken’ with a ‘certain
property’
such that ‘something happens’. For ‘every object’ with a ‘certain property’, something
happens.
Universal Quantifiers: For all, for each, for any
These are universal quantifiers and we denoted the. When ‘ ∀ ’ is used we
are looking for a
set of objects that all share the same property.
Ex:
a) There exists an x in the set S such that P(x) is true
∃ xES : ∃ P(x) is true
b) ∃ x ∀ y, (x≥ y), x, y EZ
There is an integer. That integer is greater than or equal to all integers. There exists an x
that is greater than or equal to all integers y.
QUESTION: How do we communicate statements with quantifiers?
Negation
A ~A, ¬A
T F
F T
So the negation of is ∀ , the negation of is ∃
Therefore ~∀ is ∃
Ex: An integer n divides an integer m (n|m), if m=kn, kEZ
∃
If n|m, then kEZ : m=kn
By definition n|m means m=kn, for some kEZ MATH135 week 2 Sets (ctd), Quantifiers, Nested Quantifiers, Induction Sept. 16-20, 2013
Before doing another example it is important to know the following types of compound
statements
Implication: A B OR (~A)V(B)
Converse of Implication: B A
Contrapositive: (~B)(~A) logically equivalent to implication
Negation of Implication: ~(AB) OR (A)Λ(~B)
Ex: Recall A: ∃ xES : P(x) is true
~A: ∀ xES : P(x) is false
B: ∃ x, ∀ y (x≥ y)
~B: ∀ x, ∃ y (x or ≤ instead of <
An example of a statement with ‘nested quantifiers’ is…
DEFINITION: The LIMIT of f(x) as x -> a equals L means that ε>0, ∃ δ>0
NOTE: ε is epsilon
δ is delta
0 < |x-a|ax)=L)
|x-a|ax+b)=ma+b
Let ε>0 be a real number
We want to show ∀ ε>0 ∃ δ >0 -> 0a
Define: the limit of f(x) as x->a=x->ax)=L) means that ∀ ε0 ∃ δ>0 MATH135 week 2 Sets (ctd), Quantifiers, Nested Quantifiers, Induction Sept. 16-20, 2013
0amx+b)=ma+b
Let ε>0 be a real number
We want to show ∀ε >0 ∃ δ>0 : 0im f(x)=0
Let ε>0 be a real number
We want to show that ∀ ε>0 ∃ δ>0 :
0

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