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Math Help - Find counterexamples to the following statements.

  1. #1
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    Find counterexamples to the following statements.

    I'm having trouble getting started with the following question:

    Find counterexamples to the following statements.

    A \subseteq B \Rightarrow A^c \subseteq B^c

    (A \not\subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)

    (A \subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)

    How should I make stuff up?
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  2. #2
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    Re: Find counterexamples to the following statements.

    Quote Originally Posted by maclunian View Post
    I'm having trouble getting started with the following question:

    Find counterexamples to the following statements.

    A \subseteq B \Rightarrow A^c \subseteq B^c

    (A \not\subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)

    (A \subseteq B) \wedge (B \not\subseteq C) \Rightarrow (A \not\subseteq C)
    Let \mathcal{U}=\{0,1,2,3,4,5,6,7,8,9\}
    If A=\{2,3,5,6\} then A^c=\{0,1,4,7,8,9\}.
    Now you do some of your own work.
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  3. #3
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    Re: Find counterexamples to the following statements.

    What would be an example to the first one though? How would I notate it?
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  4. #4
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    Re: Find counterexamples to the following statements.

    suppose B = {1,2,3,5,6}. is it true that A⊆B? what can you say about B^c?
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  5. #5
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    Re: Find counterexamples to the following statements.

    Quote Originally Posted by Deveno View Post
    suppose B = {1,2,3,5,6}. is it true that A⊆B?
    Well that would depend on what A = ?

    If A = {2,3,5,6} and B = {1,2,3,5,6} then it is true that A⊆B, right?
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  6. #6
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    Re: Find counterexamples to the following statements.

    assume the same A as Plato gave you.
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  7. #7
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    Re: Find counterexamples to the following statements.

    then it is true that A⊆B, right?
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  8. #8
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    Re: Find counterexamples to the following statements.

    So here's my answer for the first one:

    A \subseteq B \Rightarrow A^c \not\subseteq B^c

    because \{2,3,5,6\} \subseteq \{1,2,3,5,6\} \Rightarrow \{0,1,4,7,8,9\} \not\subseteq \{0,4,7,8,9\}

    Is this correct?
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  9. #9
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    Re: Find counterexamples to the following statements.

    Quote Originally Posted by maclunian View Post
    So here's my answer for the first one:

    A \subseteq B \Rightarrow A^c \not\subseteq B^c

    because \{2,3,5,6\} \subseteq \{1,2,3,5,6\} \Rightarrow \{0,1,4,7,8,9\} \not\subseteq \{0,4,7,8,9\}

    Is this correct?
    You tell us if it is correct or not.
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  10. #10
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    Re: Find counterexamples to the following statements.

    Quote Originally Posted by maclunian View Post
    So here's my answer for the first one:

    A \subseteq B \Rightarrow A^c \not\subseteq B^c
    whether this statement is correct or not, it is not what you were asked to show. you were asked to find a counter-example, that is, sets A and B for which:

     A\subseteq B \not\implies A^c \subseteq B^c

    so you should have a statement like:

    A \subseteq B but A^c \not \subseteq B^c,

    for some specific A and B (one example, not a general case which the counter-example "proves").
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