How is the union defined using OR?

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Discussion Overview

The discussion centers around the definition of the logical operator "OR" and its application in set theory, particularly in the context of defining the union of two sets. Participants explore the implications of circular definitions and the relationship between logical operations and set membership.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants define the union of sets A and B using the notation A∪B and express concern about the circularity of defining "OR" as true when either A or B is true or both.
  • Others argue that the definition of "OR" is simply a binary function that maps pairs of truth values to a truth value, which is false only when both inputs are false.
  • A participant suggests that the circularity of the definition may depend on how conjunction (∧) and implication (⇒) are defined.
  • Some participants assert that there is nothing circular in the definitions provided and emphasize the distinction between sets and the conditions that define them.
  • There is a mention of the truth table for "OR" as a means to define the logical operation, with some participants insisting that "OR" and disjunction are interchangeable terms.
  • One participant expresses confusion about the relevance of set theory to the question of defining "OR" and acknowledges a potential misunderstanding in framing the discussion.

Areas of Agreement / Disagreement

Participants express differing views on whether the definition of "OR" is circular. Some maintain that it is not circular, while others question the clarity of the definitions provided. The discussion remains unresolved regarding the nature of the definitions and their implications.

Contextual Notes

Participants highlight the importance of distinguishing between logical operations and set definitions, with some suggesting that the definitions could be clearer to avoid confusion. The discussion also touches on the definitions of conjunction and implication, which are not fully resolved.

Swapnil
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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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It's defined as or. As in A v B is the condition that A holds, or B holds, or both hold.
 
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!
 
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.
 
A | B | A V B
-----------------------------------
T | T | T
T | F | T
F | T | T
F | F | F
 
verty said:
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]...
 
What exactly is circular in the definition?
 
Swapnil said:
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?

Swapnil said:
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.
 
radou said:
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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  • #10
cristo said:
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.
 
  • #11
Swapnil said:
I was actually talking about the definition of OR as mentioned by DeadWolfe.

Sorry, I read the post incorrectly :blushing:
 
  • #12
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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  • #13
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
DeadWolfe said:
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?
 
  • #15
matt grime said:
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: ).
 
  • #16
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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