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

# What are some nested quantifiers? Description

##### Description: A quick review of the material covered.
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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. . .

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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.

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Negating Quantified Expressions If you want to learn more check out What do managerial accountants do?
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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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