How is the union defined using OR?

[Swapnil]

Defining OR

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?

 

[DeadWolfe]

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

 

[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!

 

[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.

 

[radou]

A | B | A V B
-----------------------------------
T | T | T
T | F | T
F | T | T
F | F | F

 

[Swapnil]

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 $$\Rightarrow$$...

 

[radou]

What exactly is circular in the definition?

 

[cristo]

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?

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.

 

[Swapnil]

What exactly is circular in the definition?

Well... nothing yet. Until you start defining $$\land$$ and $$\Rightarrow$$

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

 

[Swapnil]

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]

I was actually talking about the definition of OR as mentioned by DeadWolfe.

Sorry, I read the post incorrectly

 

[matt grime]

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.

 

[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.

 

[Swapnil]

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?

 

[Swapnil]

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. ).

 

[matt grime]

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).