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Thread: empty set

  1. #1
    Junior Member
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    empty set

    I define S as empty set:
    S = {}
    Why the Cartesian product S x A (A is not empty set) not defined?
    Is
    (2) S x S x A
    (3) A x S
    (4) S x A x S
    defined?
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  2. #2
    MHF Contributor
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    Re: empty set

    Quote Originally Posted by policer View Post
    I define S as empty set:
    S = {}
    Why the Cartesian product S x A (A is not empty set) not defined?
    Is
    (2) S x S x A
    (3) A x S
    (4) S x A x S
    defined?
    It may not be defined in your book, but other books define it. Here is how one might define the Cartesian product with the empty set:

    Given sets $A,B$, we define $A\times B = \{(a,b)|a\in A, b\in B\}$. If $A=\emptyset$, then there are no candidate choices for $a\in A$, so there are no ordered pairs in $A\times B$, making it the empty set. Similarly if $B=\emptyset$.
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