Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Unions and Intersections

  1. Feb 10, 2007 #1
    Defining OR

    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?
     
    Last edited: Feb 10, 2007
  2. jcsd
  3. Feb 10, 2007 #2
    It's defined as or. As in A v B is the condition that A holds, or B holds, or both hold.
     
  4. Feb 10, 2007 #3
    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!
     
  5. Feb 10, 2007 #4

    verty

    User Avatar
    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.
     
  6. Feb 10, 2007 #5

    radou

    User Avatar
    Homework Helper

    A | B | A V B
    -----------------------------------
    T | T | T
    T | F | T
    F | T | T
    F | F | F
     
  7. Feb 10, 2007 #6
    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]...
     
  8. Feb 10, 2007 #7

    radou

    User Avatar
    Homework Helper

    What exactly is circular in the definition?
     
  9. Feb 10, 2007 #8

    cristo

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  10. Feb 10, 2007 #9
    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]
     
    Last edited: Feb 10, 2007
  11. Feb 10, 2007 #10
    I was actually talking about the definition of OR as mentioned by DeadWolfe.
     
  12. Feb 10, 2007 #11

    cristo

    User Avatar
    Staff Emeritus
    Science Advisor

    Sorry, I read the post incorrectly :blushing:
     
  13. Feb 11, 2007 #12

    matt grime

    User Avatar
    Science Advisor
    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.
     
    Last edited: Feb 11, 2007
  14. Feb 11, 2007 #13
    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.
     
  15. Feb 11, 2007 #14
    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?
     
  16. Feb 11, 2007 #15
    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. :blushing: ).
     
  17. Feb 11, 2007 #16

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?