Unions and Intersections

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Main Question or Discussion Point

Defining OR

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?
 
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452
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It's defined as or. As in A v B is the condition that A holds, or B holds, or both hold.
 
459
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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!
 
verty
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Perhaps this is better. It is a binary function that maps 2-tuples of truth values to a truth value which is false for (0,0) and true otherwise.

Oh, perhaps this is circular.
 
radou
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A | B | A V B
-----------------------------------
T | T | T
T | F | T
F | T | T
F | F | F
 
459
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Perhaps this is better. It is a binary function that maps 2-tuples of truth values to a truth value which is false for (0,0) and true otherwise.

Oh, perhaps this is circular.
I think this is circular too.

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]...
 
radou
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What exactly is circular in the definition?
 
cristo
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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?
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!
This is not a definition of "A or B"; it is a definition of the union of the sets A and B. This is not a circular definition.
 
459
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What exactly is circular in the definition?
Well... nothing yet. Until you start defining [tex]\land[/tex] and [tex]\Rightarrow[/tex]

Notice that
[tex] p \Rightarrow q : = \lnot p \lor q[/tex]
 
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459
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This is not a definition of "A or B"; it is a definition of the union of the sets A and B. This is not a circular definition.
I was actually talking about the definition of OR as mentioned by DeadWolfe.
 
cristo
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matt grime
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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.
 
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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.
 
459
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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.
But they are the same thing! Call it 'V', or 'OR' or 'or.' It is still a logical OR.

Anyways, say that you do define v to be or. The how do you then define or?
 
459
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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.
I know that. I am just asking how the disjunction (OR) is defined. (I guess I should have never brought sets in my question. And my title was a big mistake too. :blushing: ).
 
matt grime
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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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