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COLORADO / Computer Science and Engineering / CS 2824 / What are some nested quantifiers?

What are some nested quantifiers?

What are some nested quantifiers?

Description

School: University of Colorado at Boulder
Department: Computer Science and Engineering
Course: Discrete Structures
Professor: Christian ketelson
Term: Spring 2017
Tags: DescreteMath
Cost: 50
Name: CSCI 2824 Study guide
Description: A quick review of the material covered.
Uploaded: 01/11/2017
8 Pages 130 Views 1 Unlocks
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We also discuss several other topics like What makes up a marketing plan project?

Existential Quantifier

  • ∃x P(x) ∈> "there exists a x in the proposition P(x).
  • domain must be indicated.
  • the meaning changes with the domain.
  • ∃x P(x) is False iff P(x) is False for all elements.
  • the domain of discourse is non-empty
  • ∃P(x)∈> PC x J V P (x) V. . . 

If you want to learn more check out What are the characteristics of pterosaurs?

Precedence of Quantifiers

  • The Universal and Existential Quantifiers have precedence over all the other logical operations

Local Equivalence involving Quantifiers

  • Predicates and Quantifiers are logically equivalent if and only if they have the same truth value for any predicate.

We also discuss several other topics like Who were the early founders of cultural anthropology?

Negating Quantified Expressions If you want to learn more check out What do managerial accountants do?
Don't forget about the age old question of What are the different management methods?

1) ᆨ∀x P(x) ≡ ∃x ᆨP(x)

2) ᆨ∃x Q(x) ≡ ∀x ᆨQ(x)

Don't forget about the age old question of What is the difference between interest expense and interest payable?

Section 1.5-Nested Quantifiers

        ∀x P(x)                        ∀x ∀y P(x,y)

        ↓                                ↓

        “Not nested”                     “nested”

Statement

True

False

∀x ∀y P(x,y)

∀x ∀y P(x,y)

True for all

Pairs of (x,y)

There is a pair

That make it F

∀x ∃y P(x,y)

For every x there is a y

There is an x and y

∃x ∀y P(x,y)

There is an x for which P is true for y

All x there is a y

∃x ∃y P(x,y)

∃y ∃x P(x,y)

There is a pair (x,y)

That makes P(x,y) true

Is false =

Every pair (x,y)

  • see text for conversion between symbolic statement and English.

Conjunction "and" "but " (∧)

p        q        p∧q                        - only true when both p and q are true

T        T        T                        

T        F        F                        “I have a cat and a snake”

F        T        F                                p                q

F        F        F                                false                true

                                        Both aren’t true so, p∧q is false.


Disjunction (Inclusive or) “or” (pvq)

p        q        pvq                        - false when both p and q are false

T        T        T                        

T        F        T                        “I have a cat or I don't have a snake”

F        T        T                                P                q

F        F        F                                false                false

                                        Both are false so the compound proposition is false.


Exclusive or (⊕)

p        q        p⊕q                        - true when exactly one of p and q is true.

T        T        F

T        F        T                        “I have a cat or a snake”

F        T        T                                p                q

F        F        T                                false                true

                                        The compound proposition is true.


Section 2.1: Application of Propositional logic

Translating English Sentences

  • Use logical operations word representation

“I have a snack, but I don’t have a cat”

        p                        ᆨp

                p ∧ ᆨp

System Specifications

  • Should be consistent, not containing conflicting requirements

Logical Circuits

        inverter                                and gate                        or gate

Gates to statement

* to go from the statement to the gate representation just break the statement into parts.

Conditionals (→)

“If p, then q” is false when p is true and q is false and true otherwise.

     ↓             ⤷ conclusion

hypothesis

  • Obligation / rule / contract
  • See page 6 of the text for word representations.

p        q        p → q                “if I have a snake, then I have a cat”.

T        T        T

T        F        F

F        T        T                p = true                        p → q = false

F        F        T                q = false

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