# Unions and Intersections

by Swapnil
Tags: intersections, unions
 P: 460 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?
 P: 461 It's defined as or. As in A v B is the condition that A holds, or B holds, or both hold.
 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!
 HW Helper P: 1,808 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.
 HW Helper P: 3,220 A | B | A V B ----------------------------------- T | T | T T | F | T F | T | T F | F | F
P: 460
 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 $$\Rightarrow$$...
 HW Helper P: 3,220 What exactly is circular in the definition?
Mentor
P: 8,317
 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.
P: 460
 Quote by radou What exactly is circular in the definition?
Well... nothing yet. Until you start defining $$\land$$ and $$\Rightarrow$$

Notice that
$$p \Rightarrow q : = \lnot p \lor q$$
P: 460
 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
P: 8,317
 Quote by Swapnil I was actually talking about the definition of OR as mentioned by DeadWolfe.
Sorry, I read the post incorrectly
 Sci Advisor HW Helper 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.
 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.
P: 460
 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?
P: 460
 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. ).
 Sci Advisor HW Helper 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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