The document discusses predicate logic and quantification. It introduces predicates, universal and existential quantifiers, and provides examples of their usage. It also discusses uniqueness quantification, nested quantification, and the importance of quantifier order. Negation of quantified expressions is explained. The concept of a limit of a function is introduced and its negation is derived.
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
Day 3 of Free Intuitive Calculus Course: Limits by FactoringPablo Antuna
Today we focus on limits by factoring. We solve limits by factoring and cancelling. This is one of the basic techniques for solving limits. We talk about the idea behind this technique and we solve some examples step by step.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
In this second day we solve the most basic limits we could find, like the limit of a constant. Then we find the limit of the sum, the product and the quotient of two functions. We solve two simple examples.
Day 3 of Free Intuitive Calculus Course: Limits by FactoringPablo Antuna
Today we focus on limits by factoring. We solve limits by factoring and cancelling. This is one of the basic techniques for solving limits. We talk about the idea behind this technique and we solve some examples step by step.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
In this second day we solve the most basic limits we could find, like the limit of a constant. Then we find the limit of the sum, the product and the quotient of two functions. We solve two simple examples.
1. Lecture 3
• p=“All humans are mortal.”
• q=“Hypatia is a human.”
• Does it follow that “Hypatia is mortal?”
• In propositional logic these would be two
unrelated propositions
• We need a language to encode sets and
variables (e.g., the set of humans and the
element “Hypatia”)
2. Predicate Logic (First
order Logic)
• Predicate P(x,y,z) is a statement involving a
variable, e.g., x+y<z
• Universal quantifier xP (x)
• For all x (in the domain), P(x) x DP (x)
• Existential quantifier xP (x)
• There is an element x (in the domain)
such that P(x) x DP (x)
3. Example
• Let “x - y = z” be denoted by Q(x, y, z).
Find these truth values:
• Q(2,-1,3)
• Solution: T
• Q(3,4,7)
• Solution: F
• Q(x, 3, z)
• Solution: Not a Proposition
4. Set Notation
Z = {. . . , 2, 1, 0, 1, 2, . . . } set of integers
N = {x Z : x 0} set of natural numbers
Z = {x Z : x > 0} set of positive integers
+
Q = {p/q : p Z, q Z {0}}set of rational numbers
R = the set of real numbers
5. Examples
x Z(x > 0)
x Z (x > 0)
+
x Z (x < 0)
+
x Z (x is even)
6. Examples
x Z(x > 0) is false
x Z (x > 0) is true
+
x Z (x < 0) is false
+
x Z (x is even) is true
7. Propositional Logic is
not Enough
• “All humans are mortal.”
• “Hypatia is a human.”
xHuman(x) M ortal(x)
Human(Hypatia)
= M ortal(Hypatia)
8. Uniqueness Quantifier
• !x U (P (x)) means that P(x) is true for
one and only one x in the universe of
discourse.
• “There is a unique x such that P(x).”
• “There is one and only one x such that
P(x)”
9. Uniqueness Quantifier
• Examples: !x Z(x + 1 = 0) is true
!x Z(x > 0) is false
• The uniqueness quantifier is not really
needed as the restriction that there is a
unique x such that P(x) can be expressed
as:
x(P (x) y(P (y) (y = x)))
10. Are These Negations
Correct?
• Every animal wags its tail when it is happy
• No animal wags its tail when it is happy.
• There is an animal that wags its tail when
happy
• There is an animal that does not wag its
tail when happy
11. Correct Negations
• Every animal wags its tail when it is happy
• There is an animal that does not wag its
tail when it is happy
• There is an animal that wags its tail when
happy
• All animals do not wag their tail when
happy
14. Nested Quantifiers
y R x R:x+y =0
• There is a y such that for all x, x+y=0
• Is this true?
15. y R x R:x+y =0
• There is a y such that for all x, x+y=0
• False!
• The correct proposition is the following:
y R x R:x+y =0
16. The Order of
Quantifiers is Important
y xP (x, y) is true = x yP (x, y) is true
• The converse might not be true!
x yP (x, y) is true = y xP (x, y) is true
18. The Negation is True
(so original is false)
x
¬ x>0 y>0 =1
y
x
x>0 y>0 =1
y
e.g., x = 3, y = 4
19. The Negation is True
(so the original is false)
x
¬ x>0 y>0 =1
y
x
x>0 y>0 =1
y
e.g., y = 2x
20. Example:
Limit of a Function
lim f (x) = L :
x a
>0 > 0 x(0 < |x a| < |f (x) L| <
• Can be considered as a game (or challenge)
• You give me any > 0
• I guarantee you that I can find an interval
0 < |x a| <
• Such that for all values of x in that interval,
the distance from f(x) to L is smaller than
21. Negating Limit
Definition
lim f (x) = L
x a
¬( > 0 > 0 x(0 < |x a| < |f (x) L| < ))
> 0 > 0 x¬(0 < |x a| < |f (x) L| < )
> 0 > 0 x(0 < |x a| < |f (x) L| )
The last step uses the equivalence ¬(p→q) ≡ p∧¬q