Unions and Intersections

Swapnil

Given set A and B, the union is defined as

[tex]A\cup B := \{x | x \; \epsilon A \lor x \; \epsilon \; B \}[/tex]

But how is [tex]\lor[/tex] 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

I think this is circular too.

Correct me if I am wrong. You define OR as a function [tex]f: (x,y) \to z[/tex] where [tex] x,y,z \; \epsilon \; \{0, 1\}[/tex] satisfying the following property:

[tex](x,y) = (0,0) \Rightarrow z = 0 \land (x,y) \neq (0,0) \Rightarrow z = 1 [/tex]

I guess the circularity of this definition depends on how you define [tex]\land[/tex] and [tex]\Rightarrow[/tex]...

radou

What exactly is circular in the definition?

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.

Swapnil

Well... nothing yet. Until you start defining [tex]\land[/tex] and [tex]\Rightarrow[/tex]

Notice that
[tex] p \Rightarrow q : = \lnot p \lor q[/tex]

Swapnil

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

cristo

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

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

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).