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Math Help - increasing functions

  1. #1
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    increasing functions

    Let P ({0,1}) denote the power set of the set {0,1}, We say that a function
    f:P({0,1}) to {1,2,...,5} is increasing function if for any
    A,B belongs toP({0,1}) we have f (A) < f(B) whenever A is a proper subset of
    B . How many such increasing functions are injective and how many are
    not.
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  2. #2
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    Quote Originally Posted by makenqau View Post
    Let P ({0,1}) denote the power set of the set {0,1}, We say that a function
    f:P({0,1}) to {1,2,...,5} is increasing function if for any
    A,B belongs toP({0,1}) we have f (A) < f(B) whenever A is a proper subset of
    B . How many such increasing functions are injective and how many are
    not.
    Say that A<B or B<A then f(A) < f(B) or f(B) < f(A).
    Therefore, \emptyset, \{ 0,1\}, \{ 0\} must be mapped into distinct elements and \emptyset, \{0,1\}, \{ 1\} must be mapped into distinct elements if f is increasing.
    However, since \{ 0 \} \not < \{ 1 \} or \{ 1 \} \not < \{ 0 \} it means we cannot conclude that f( \{ 0 \} ) \not = f(\{1\}) if f is increaing.

    Based on the above reasoning we have the following possibilities:
    1. \emptyset \mapsto 1 , \{ 0 \} \mapsto 2, \{ 1 \} \mapsto 3, \{0,1\} \mapsto 4,5
    2. \emptyset \mapsto 1, \{ 0\} \mapsto 3, \{ 1 \}\mapsto 2, \{ 0,1 \} \mapsto 4,5
    3. \emptyset \mapsto 2, \{ 0 \}\mapsto 3, \{ 1 \} \mapsto 4, \{0,1\} \mapsto 5
    4. \emptyset \mapsto 2, \{ 0 \} \mapsto 4, \{ 1 \} \mapsto 3, \{ 0,1\} \mapsto 5


    I get six.
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