Monday, October 5, 2009
song
Akon ft. Snoop Dogg - I Wanna Love You lyrics (Akon:) Convict...Music...and you know we up front. (Chorus: Akon) I see you windin' and grindin' up on the floor I know you see me lookin' at you and you already know I wanna love you, you already know I wanna love you, you already know (Snoop Dogg) Money in the air as mo' fell Grab you by your coattail, take you to the motel, ho sale Don't tell, wont tell, baby say "I don't talk, Dogg unless you told on me" - oh well Take a picture wit me, what the flick gon' do Baby stick to me and I'ma stick on you If you pick me then I'ma pick on you D-o-double g and I'm here to put this d*** on you I'm stuck on p**** and your's is right Rip ridin' the poles and them doors is tight And I'ma get me a shot 'fo the end of the night Cuz p**** is p**** and baby you're p**** for life (Chorus: Akon) I see you windin' and grindin' up on the floor I know you see me lookin' at you and you already know I wanna love you, you already know I wanna love you, you already know (Akon) Shorty I can see you ain't lonely Handful of n***** and they all got cheese See you lookin' at me now what its gon' be Just another tease far as I can see Trynna get you up out this club if it means spendin' a couple dubs Throwin' bout 30 stacks in the back make it rain like that cuz I'm far from a scrub And you know my pedigree, ex-deala use to move phetamines Girl I spend money like it don't mean nuthin' and besides I got a thing for you. (Chorus: Akon) I see you windin' and grindin' up on the floor, I know you see me lookin' at you and you already know I wanna love you, you already know I wanna love you, you already know girl (Snoop Dogg) Mobbin' through the club and I'm low pressin' I'm sittin' in the back in the smoker's section (just smokin') Birds eye, I got a clear view You can't see me, but I can see you (baby I see you) -mm It's cool, we jet, the mood is set, your p* is wet You're rubbin' your back and touchin' your neck Your body is movin', you humpin' and jumpin' Your t****** is bouncin', you smilin' and grinnin' and lookin' at me (Akon) Girl and while your looking at me I'm ready to hit the caddy Right up on the patio move the patty to the caddy Baby you got a phatty, the type I like to marry Wantin' to just give you everything and that's kinda scary Cuz I'm lovin the way you shake your ass Bouncin', got me tippin' my glass Lil' mully don't get caught up too fast But I got a thing for you (Chorus: Akon) I see you windin' and grindin' up on the floor, I know you see me lookin' at you and you already know I wanna love you, you already know I wanna love you, you already know (Chorus: Akon) I see you windin' and grindin' up on the floor, I know you see me lookin' at you and you already know I wanna love you, you already know I wanna love you, you already know Girl...
demorgans law
[edit] Formal definition
In propositional calculus form:
where:
is the negation operator (jump)
is the conjunction operator (AND)
is the disjunction operator (OR)
means logically equivalent (if and only if)
In set theory and Boolean algebra, it is often stated as "Union and intersection interchange under complementation."[1]:
where:
is the negation of A, the overline is written above the terms to be negated
is the intersection operator (AND)
is the union operator (OR)
The generalized form is:
In set notation, De Morgan's law can be remembered using the mnemonic "break the line, change the sign".[2]
[edit] History
The law is named after Augustus De Morgan (1806–1871)[3] who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraisation of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Although a similar observation was made by Aristotle and was known to Greek and Medieval logicians [4] (in the 14th century William of Ockham wrote down the words that would result by reading the laws out[5]), De Morgan is given credit for stating the laws formally and incorporating them in to the language of logic. De Morgan's Laws can be proved easily, and may even seem trivial.[6] Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.
[edit] Informal proof
De Morgan's theorem may be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula.
[edit] Negation of a disjunction
In the case of its application to a disjunction, consider the following claim: it is false that either A or B is true, which is written as:
In that it has been established that neither A nor B is true, then it must follow that A is not true and B is not true; If either A or B were true, then the disjunction of A and B would be true, making its negation false.
Working in the opposite direction with the same type of problem, consider this claim:
This claim asserts that A is false and B is false (or "not A" and "not B" are true). Knowing this, a disjunction of A and B would be false, also. However the negation of said disjunction would yield a true result that is logically equivalent to the original claim. Presented in English, this would follow the logic that "Since two things are false, it's also false that either of them are true."
[edit] Negation of a conjunction
The application of De Morgan's theorem to a conjunction is very similar to its application to a disjunction both in form and rationale. Consider the following claim: It is false that A and B are both true, which is written as:
In order for this claim to be true, either or both of A or B must be false, in that if they both were true, then the conjunction of A and B would be true, making its negation false. So, the original claim may be translated as "Either A is false or B is false", or "Either not A is true or not B is true".
Presented in English, this would follow the logic that "Since it is false that two things together are true, at least one of them must be false."
[edit] Formal proof
if and only if and .
for arbitrary x:
inclusion:
or
or
Therefore
inclusion:
or
or
Therefore
and therefore Q.E.D.
can be shown using a similar method.
[edit] Extensions
In extensions of classical propositional logic, the duality still holds (that is, to any logical operator we can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important property of logics based on classical logic, namely the existence of negation normal forms: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is a prerequisite for finding the conjunctive normal form and disjunctive normal form of a formula. Computer programmers use them to change a complicated statement like IF ... AND (... OR ...) THEN ... into its opposite. They are also often useful in computations in elementary probability theory.
Let us define the dual of any propositional operator P(p, q, ...) depending on elementary propositions p, q, ... to be the operator Pd defined by
This idea can be generalised to quantifiers, so for example the universal quantifier and existential quantifier are duals:
To relate these quantifier dualities to the De Morgan laws, set up a model with some small number of elements in its domain D, such as
D = {a, b, c}.
Then
and
But, using De Morgan's laws,
and
verifying the quantifier dualities in the model.
Then, the quantifier dualities can be extended further to modal logic, relating the box ("necessarily") and diamond ("possibly") operators:
In its application to the alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators to the quantification can be understood by setting up models using Kripke semantics.
[edit] See also
List of Boolean algebra topics
[edit] References
^ Boolean Algebra By R. L. Goodstein. ISBN 0486458946
^ 2000 Solved Problems in Digital Electronics By S. P. Bali Superman
^ DeMorgan’s Theorems at mtsu.edu
^ Bocheński's History of Formal Logic
^ William of Ockham, Summa Logicae, part II, sections 32 & 33.
^ Augustus De Morgan (1806 -1871) by Robert H. Orr
In propositional calculus form:
where:
is the negation operator (jump)
is the conjunction operator (AND)
is the disjunction operator (OR)
means logically equivalent (if and only if)
In set theory and Boolean algebra, it is often stated as "Union and intersection interchange under complementation."[1]:
where:
is the negation of A, the overline is written above the terms to be negated
is the intersection operator (AND)
is the union operator (OR)
The generalized form is:
In set notation, De Morgan's law can be remembered using the mnemonic "break the line, change the sign".[2]
[edit] History
The law is named after Augustus De Morgan (1806–1871)[3] who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraisation of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Although a similar observation was made by Aristotle and was known to Greek and Medieval logicians [4] (in the 14th century William of Ockham wrote down the words that would result by reading the laws out[5]), De Morgan is given credit for stating the laws formally and incorporating them in to the language of logic. De Morgan's Laws can be proved easily, and may even seem trivial.[6] Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.
[edit] Informal proof
De Morgan's theorem may be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula.
[edit] Negation of a disjunction
In the case of its application to a disjunction, consider the following claim: it is false that either A or B is true, which is written as:
In that it has been established that neither A nor B is true, then it must follow that A is not true and B is not true; If either A or B were true, then the disjunction of A and B would be true, making its negation false.
Working in the opposite direction with the same type of problem, consider this claim:
This claim asserts that A is false and B is false (or "not A" and "not B" are true). Knowing this, a disjunction of A and B would be false, also. However the negation of said disjunction would yield a true result that is logically equivalent to the original claim. Presented in English, this would follow the logic that "Since two things are false, it's also false that either of them are true."
[edit] Negation of a conjunction
The application of De Morgan's theorem to a conjunction is very similar to its application to a disjunction both in form and rationale. Consider the following claim: It is false that A and B are both true, which is written as:
In order for this claim to be true, either or both of A or B must be false, in that if they both were true, then the conjunction of A and B would be true, making its negation false. So, the original claim may be translated as "Either A is false or B is false", or "Either not A is true or not B is true".
Presented in English, this would follow the logic that "Since it is false that two things together are true, at least one of them must be false."
[edit] Formal proof
if and only if and .
for arbitrary x:
inclusion:
or
or
Therefore
inclusion:
or
or
Therefore
and therefore Q.E.D.
can be shown using a similar method.
[edit] Extensions
In extensions of classical propositional logic, the duality still holds (that is, to any logical operator we can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important property of logics based on classical logic, namely the existence of negation normal forms: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is a prerequisite for finding the conjunctive normal form and disjunctive normal form of a formula. Computer programmers use them to change a complicated statement like IF ... AND (... OR ...) THEN ... into its opposite. They are also often useful in computations in elementary probability theory.
Let us define the dual of any propositional operator P(p, q, ...) depending on elementary propositions p, q, ... to be the operator Pd defined by
This idea can be generalised to quantifiers, so for example the universal quantifier and existential quantifier are duals:
To relate these quantifier dualities to the De Morgan laws, set up a model with some small number of elements in its domain D, such as
D = {a, b, c}.
Then
and
But, using De Morgan's laws,
and
verifying the quantifier dualities in the model.
Then, the quantifier dualities can be extended further to modal logic, relating the box ("necessarily") and diamond ("possibly") operators:
In its application to the alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators to the quantification can be understood by setting up models using Kripke semantics.
[edit] See also
List of Boolean algebra topics
[edit] References
^ Boolean Algebra By R. L. Goodstein. ISBN 0486458946
^ 2000 Solved Problems in Digital Electronics By S. P. Bali Superman
^ DeMorgan’s Theorems at mtsu.edu
^ Bocheński's History of Formal Logic
^ William of Ockham, Summa Logicae, part II, sections 32 & 33.
^ Augustus De Morgan (1806 -1871) by Robert H. Orr
demorgans law
[edit] Formal definition
In propositional calculus form:
where:
is the negation operator (jump)
is the conjunction operator (AND)
is the disjunction operator (OR)
means logically equivalent (if and only if)
In set theory and Boolean algebra, it is often stated as "Union and intersection interchange under complementation."[1]:
where:
is the negation of A, the overline is written above the terms to be negated
is the intersection operator (AND)
is the union operator (OR)
The generalized form is:
In set notation, De Morgan's law can be remembered using the mnemonic "break the line, change the sign".[2]
[edit] History
The law is named after Augustus De Morgan (1806–1871)[3] who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraisation of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Although a similar observation was made by Aristotle and was known to Greek and Medieval logicians [4] (in the 14th century William of Ockham wrote down the words that would result by reading the laws out[5]), De Morgan is given credit for stating the laws formally and incorporating them in to the language of logic. De Morgan's Laws can be proved easily, and may even seem trivial.[6] Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.
[edit] Informal proof
De Morgan's theorem may be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula.
[edit] Negation of a disjunction
In the case of its application to a disjunction, consider the following claim: it is false that either A or B is true, which is written as:
In that it has been established that neither A nor B is true, then it must follow that A is not true and B is not true; If either A or B were true, then the disjunction of A and B would be true, making its negation false.
Working in the opposite direction with the same type of problem, consider this claim:
This claim asserts that A is false and B is false (or "not A" and "not B" are true). Knowing this, a disjunction of A and B would be false, also. However the negation of said disjunction would yield a true result that is logically equivalent to the original claim. Presented in English, this would follow the logic that "Since two things are false, it's also false that either of them are true."
[edit] Negation of a conjunction
The application of De Morgan's theorem to a conjunction is very similar to its application to a disjunction both in form and rationale. Consider the following claim: It is false that A and B are both true, which is written as:
In order for this claim to be true, either or both of A or B must be false, in that if they both were true, then the conjunction of A and B would be true, making its negation false. So, the original claim may be translated as "Either A is false or B is false", or "Either not A is true or not B is true".
Presented in English, this would follow the logic that "Since it is false that two things together are true, at least one of them must be false."
[edit] Formal proof
if and only if and .
for arbitrary x:
inclusion:
or
or
Therefore
inclusion:
or
or
Therefore
and therefore Q.E.D.
can be shown using a similar method.
[edit] Extensions
In extensions of classical propositional logic, the duality still holds (that is, to any logical operator we can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important property of logics based on classical logic, namely the existence of negation normal forms: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is a prerequisite for finding the conjunctive normal form and disjunctive normal form of a formula. Computer programmers use them to change a complicated statement like IF ... AND (... OR ...) THEN ... into its opposite. They are also often useful in computations in elementary probability theory.
Let us define the dual of any propositional operator P(p, q, ...) depending on elementary propositions p, q, ... to be the operator Pd defined by
This idea can be generalised to quantifiers, so for example the universal quantifier and existential quantifier are duals:
To relate these quantifier dualities to the De Morgan laws, set up a model with some small number of elements in its domain D, such as
D = {a, b, c}.
Then
and
But, using De Morgan's laws,
and
verifying the quantifier dualities in the model.
Then, the quantifier dualities can be extended further to modal logic, relating the box ("necessarily") and diamond ("possibly") operators:
In its application to the alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators to the quantification can be understood by setting up models using Kripke semantics.
[edit] See also
List of Boolean algebra topics
[edit] References
^ Boolean Algebra By R. L. Goodstein. ISBN 0486458946
^ 2000 Solved Problems in Digital Electronics By S. P. Bali Superman
^ DeMorgan’s Theorems at mtsu.edu
^ Bocheński's History of Formal Logic
^ William of Ockham, Summa Logicae, part II, sections 32 & 33.
^ Augustus De Morgan (1806 -1871) by Robert H. Orr
In propositional calculus form:
where:
is the negation operator (jump)
is the conjunction operator (AND)
is the disjunction operator (OR)
means logically equivalent (if and only if)
In set theory and Boolean algebra, it is often stated as "Union and intersection interchange under complementation."[1]:
where:
is the negation of A, the overline is written above the terms to be negated
is the intersection operator (AND)
is the union operator (OR)
The generalized form is:
In set notation, De Morgan's law can be remembered using the mnemonic "break the line, change the sign".[2]
[edit] History
The law is named after Augustus De Morgan (1806–1871)[3] who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraisation of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Although a similar observation was made by Aristotle and was known to Greek and Medieval logicians [4] (in the 14th century William of Ockham wrote down the words that would result by reading the laws out[5]), De Morgan is given credit for stating the laws formally and incorporating them in to the language of logic. De Morgan's Laws can be proved easily, and may even seem trivial.[6] Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.
[edit] Informal proof
De Morgan's theorem may be applied to the negation of a disjunction or the negation of a conjunction in all or part of a formula.
[edit] Negation of a disjunction
In the case of its application to a disjunction, consider the following claim: it is false that either A or B is true, which is written as:
In that it has been established that neither A nor B is true, then it must follow that A is not true and B is not true; If either A or B were true, then the disjunction of A and B would be true, making its negation false.
Working in the opposite direction with the same type of problem, consider this claim:
This claim asserts that A is false and B is false (or "not A" and "not B" are true). Knowing this, a disjunction of A and B would be false, also. However the negation of said disjunction would yield a true result that is logically equivalent to the original claim. Presented in English, this would follow the logic that "Since two things are false, it's also false that either of them are true."
[edit] Negation of a conjunction
The application of De Morgan's theorem to a conjunction is very similar to its application to a disjunction both in form and rationale. Consider the following claim: It is false that A and B are both true, which is written as:
In order for this claim to be true, either or both of A or B must be false, in that if they both were true, then the conjunction of A and B would be true, making its negation false. So, the original claim may be translated as "Either A is false or B is false", or "Either not A is true or not B is true".
Presented in English, this would follow the logic that "Since it is false that two things together are true, at least one of them must be false."
[edit] Formal proof
if and only if and .
for arbitrary x:
inclusion:
or
or
Therefore
inclusion:
or
or
Therefore
and therefore Q.E.D.
can be shown using a similar method.
[edit] Extensions
In extensions of classical propositional logic, the duality still holds (that is, to any logical operator we can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important property of logics based on classical logic, namely the existence of negation normal forms: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is a prerequisite for finding the conjunctive normal form and disjunctive normal form of a formula. Computer programmers use them to change a complicated statement like IF ... AND (... OR ...) THEN ... into its opposite. They are also often useful in computations in elementary probability theory.
Let us define the dual of any propositional operator P(p, q, ...) depending on elementary propositions p, q, ... to be the operator Pd defined by
This idea can be generalised to quantifiers, so for example the universal quantifier and existential quantifier are duals:
To relate these quantifier dualities to the De Morgan laws, set up a model with some small number of elements in its domain D, such as
D = {a, b, c}.
Then
and
But, using De Morgan's laws,
and
verifying the quantifier dualities in the model.
Then, the quantifier dualities can be extended further to modal logic, relating the box ("necessarily") and diamond ("possibly") operators:
In its application to the alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators to the quantification can be understood by setting up models using Kripke semantics.
[edit] See also
List of Boolean algebra topics
[edit] References
^ Boolean Algebra By R. L. Goodstein. ISBN 0486458946
^ 2000 Solved Problems in Digital Electronics By S. P. Bali Superman
^ DeMorgan’s Theorems at mtsu.edu
^ Bocheński's History of Formal Logic
^ William of Ockham, Summa Logicae, part II, sections 32 & 33.
^ Augustus De Morgan (1806 -1871) by Robert H. Orr
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