# Unions and Intersections

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?

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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.

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!

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

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 $$\Rightarrow$$...

Homework Helper
What exactly is circular in the definition?

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.

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

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

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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.

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

Sorry, I read the post incorrectly

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.

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. ).