
#1
Mar612, 10:53 AM

P: 88

This can considered as a logic question , when I say for example there exist x such that ... . Mustn't I define some set from which x belong .
In a book of set theory it defined a binary relation as following : A set R is a binary relation if [itex](\forall x \in R)(\exists x)(\exists y)(z=(x,y))[/itex] The way I understand [itex]\exists x[/itex] is as following , as he is referring to any set x which exist , So we must consider some Set containing all sets , Such set doesn't exist . So what set must be considered , how must I understand this I know that we didn't mention the universal set if it is clear from context . Here , there is no Universal set . we want x to be arbitrary . Thanks 



#2
Mar612, 06:37 PM

Sci Advisor
P: 5,935

It looks like the author is implicitly assuming x and y belong to some set (R?). Also the statement starts with for all x in R  shouldn't it be for all z in R?




#3
Mar712, 09:36 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,879





#4
Mar712, 01:11 PM

P: 159

What is meant by that ??You seem to be confusing this with the axiom of specification, which says that given a set A, you can define a set B as the set of all [itex]x \in A[/itex] such that [itex]\varphi(x)[/itex]. 



#5
Mar712, 04:19 PM

P: 88

So how mus I understand it 



#6
Mar712, 05:34 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,879

Every z in the relation consists of a pair of real numbers.




#7
Mar712, 06:09 PM

P: 88





#8
Mar812, 04:37 PM

P: 27





#9
Mar912, 05:18 AM

P: 88





#10
Mar912, 12:54 PM

P: 27

It should mean that out of a certain set ordinarily called "the domain" you have created a subset consisting of all objects ,x, satisfying the condition "P". Note: You must distinguish between "x" as any object of a set and "x" considered as the set! If x is any element of the set x then x contains itself as an element and there is a certain axiom in ZFC forbidding just that. (Since the axiom is independent of the other axioms, then you could use ZFC with the axiom replaced with its negation resulting in a set theory as consistent as ZFC is.) 2 When you say there exists some x such that P(x) holds, then you claim that your subset of the domain is not empty. 



#11
Mar912, 03:24 PM

P: 88





#12
Mar912, 04:06 PM

P: 27





#13
Mar1012, 05:33 AM

P: 88

In the definition above it and and some of axioms of set theory for example Axiom of Extensionality we begin by saying for all x and y , .... . So what " for all " Exactly mean. 



#14
Mar1012, 06:05 AM

P: 88





#15
Mar1012, 07:08 AM

P: 27

I dont mind looking at basic definitions... Do you have access to them? Some authors does not make the foundation of their theory clear and complete. Make a list of the statements you find unclear. Are you really bothered about the meaning of "for all x..." and "there exists an x..."? The expressions are called "quantifiers" and belong to "Predicate Logic" they are perhaps assumed to already be defined and understood in the theory you are studying. "for all x..." is called the universal quantifier and it affects a variable found in a sentence function: P(x) ... "x" is a variable so P(x) is a statement function not a statement...you get a statement if you replace x with an "individual constant" say ,a, ...then P(a) is (because of the quantifier) a true sentence. An "interpretation" of the logical language used is some non empty set called "the Domain" it contains all sentences all predicates and all constants: "for all x..." now means: for each object x in the domain... Having the universal we define the existential: "there is an x such that p(x)" means "not for all x : not p(x)" (Note that the domain is assumed not exhibited! It might be infinite so we cant exhibit it.) This is a very abbreviated view...does it make any sense to you? 


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