# Unions and Intersections

## Main Question or Discussion Point

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?

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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!

verty
Homework Helper
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.

Homework Helper
A | B | A V B
-----------------------------------
T | T | T
T | F | T
F | T | T
F | F | F

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 $$\Rightarrow$$...

Homework Helper
What exactly is circular in the definition?

cristo
Staff Emeritus
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.

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

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

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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
Staff Emeritus
I was actually talking about the definition of OR as mentioned by DeadWolfe.
Sorry, I read the post incorrectly

matt grime
Homework Helper
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.

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?

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