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Unions and Intersections 
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#1
Feb1007, 07:03 PM

P: 460

Given set A and B, the union is defined as
[tex]A\cup B := \{x  x \; \epsilon A \lor x \; \epsilon \; B \}[/tex] But how is [tex]\lor[/tex] defined? 


#2
Feb1007, 07:10 PM

P: 461

It's defined as or. As in A v B is the condition that A holds, or B holds, or both hold.



#3
Feb1007, 07:55 PM

P: 460

But isn't that circular definition? You are defining A OR B as true when either A is true OR B is true OR both are true!



#4
Feb1007, 07:59 PM

HW Helper
P: 1,755

Unions and Intersections
Perhaps this is better. It is a binary function that maps 2tuples of truth values to a truth value which is false for (0,0) and true otherwise.
Oh, perhaps this is circular. 


#5
Feb1007, 08:39 PM

HW Helper
P: 3,224

A  B  A V B
 T  T  T T  F  T F  T  T F  F  F 


#6
Feb1007, 08:50 PM

P: 460

Correct me if I am wrong. You define OR as a function [tex]f: (x,y) \to z[/tex] where [tex] x,y,z \; \epsilon \; \{0, 1\}[/tex] satisfying the following property: [tex](x,y) = (0,0) \Rightarrow z = 0 \land (x,y) \neq (0,0) \Rightarrow z = 1 [/tex] I guess the circularity of this definition depends on how you define [tex]\land[/tex] and [tex]\Rightarrow[/tex]... 


#8
Feb1007, 09:02 PM

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#9
Feb1007, 09:27 PM

P: 460

Notice that [tex] p \Rightarrow q : = \lnot p \lor q[/tex] 


#10
Feb1007, 09:29 PM

P: 460




#12
Feb1107, 03:02 AM

Sci Advisor
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P: 9,396

There is nothing at all 'circular' in any of these definitions. It would have been better written as
(x in A)v(x in B) to avoid confusion (his A and B are not your A and B). What on earth do you think the definition of logical OR is if not what was given? V is just another symbol for logical OR. Do'nt confuse sets with conditions that define the sets: the defining condition for a union of two sets is the disjunction (OR) of the individual conditions. 


#13
Feb1107, 03:06 PM

P: 461

How on earth is my definition is circular. I said that v is defined to be or. Not that or is defined to be or. Pay attention.



#14
Feb1107, 05:48 PM

P: 460

Anyways, say that you do define v to be or. The how do you then define or? 


#15
Feb1107, 05:52 PM

P: 460




#16
Feb1107, 05:59 PM

Sci Advisor
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P: 9,396

Look at the (expletive deleted) truth table. That is how OR and DISJUNCTION are defined (they are after all just different names for the same thing).



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