## Unions and Intersections

Given set A and B, the union is defined as

$$A\cup B := \{x | x \; \epsilon A \lor x \; \epsilon \; B \}$$

But how is $$\lor$$ 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!

## Unions and Intersections

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.
 Recognitions: Homework Help A | B | A V B ----------------------------------- T | T | T T | F | T F | T | T F | F | F

 Quote by verty 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 $$f: (x,y) \to z$$ where $$x,y,z \; \epsilon \; \{0, 1\}$$ satisfying the following property:

$$(x,y) = (0,0) \Rightarrow z = 0 \land (x,y) \neq (0,0) \Rightarrow z = 1$$

I guess the circularity of this definition depends on how you define $$\land$$ and $$\Rightarrow$$...
 Recognitions: Homework Help What exactly is circular in the definition?

Mentor
 Quote by Swapnil Given set A and B, the union is defined as $$A\cup B := \{x | x \; \epsilon A \lor x \; \epsilon \; B \}$$ But how is $$\lor$$ defined?
 Quote by Swapnil 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.

 Quote by radou What exactly is circular in the definition?
Well... nothing yet. Until you start defining $$\land$$ and $$\Rightarrow$$

Notice that
$$p \Rightarrow q : = \lnot p \lor q$$

 Quote by cristo 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.

Mentor
 Quote by Swapnil I was actually talking about the definition of OR as mentioned by DeadWolfe.
Sorry, I read the post incorrectly
 Recognitions: Homework Help Science Advisor 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.
 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.

 Quote by DeadWolfe 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?

 Quote by matt grime 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. ).
 Recognitions: Homework Help Science Advisor 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).