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  1. #1
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    Discrete Math - HELP

    this is the question:

    If you voted Obama then you have a proud Mama."


    John did not vote Obama, and has a proud Mama.
    Joe voted Obama, and has a proud Mama.
    Sarah did not vote Obama, and does not have a proud Mama.
    Decide if the above statement is true for each of these people.
    Do the same for the negation, the converse, and the contrapositive.

    how can i slove it??
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  2. #2
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    Hello, tukilala!

    I'll get you started on this one . . .


    If you voted Obama, then you have a proud Mama.

    (a) John did not vote Obama and has a proud Mama.
    (b) Joe voted Obama and has a proud Mama.
    (c) Sarah did not vote Obama and does not have a proud Mama.

    [1] Decide if the above statement is true for each of these people.

    [2] Do the same for the negation, the converse, and the contrapositive.

    \text{We have: }\:\text{If }\underbrace{\text{you voted Obama}}_p\text{, then }\underbrace{\text{you have a proud Mama.}}_q

    So we have: . p \to q\;\text{ which is equivalent to }\sim p \vee q


    (a) John did not vote Obama and has a proud Mama: . \sim p \wedge q . . . True

    (b) Joe voted Obama and has a proud Mama: .  p \wedge q . . . True

    (c) Sarah did not vote Obama and does not have a proud Mama: . \sim p\: \wedge \sim q . . . True

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  3. #3
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    Quote Originally Posted by tukilala View Post
    this is the question:



    If you voted Obama then you have a proud Mama."




    John did not vote Obama, and has a proud Mama.
    Joe voted Obama, and has a proud Mama.
    Sarah did not vote Obama, and does not have a proud Mama.
    Decide if the above statement is true for each of these people.
    Do the same for the negation, the converse, and the contrapositive.



    how can i slove it??


    The statement :


    ...."if you voted Obama then you have a good Mama "................

    is a conditional one .................................................. ............1

    Let us put : you vote Obama=p .......and you have a good Mama=q


    For the case where:


    john did not vote Obama, and has a proud Mama,


    p is false and q is true hence according to the definition of the conditional .........................(1) is true



    For the next case :.........p is true and q is true hence:


    ......................(1) is true.


    For the next case :............p is false and q is false hence:


    .........................(1) is true.


    Now the negation of (1) is ~(~pvq) <====> (by De Morgan) p^ ~q ,since p---->q is equivalent to ~pvq.............................................. ......................2



    The new statement now is not a conditional anymore but a conjunct ,hence using the definition of a conjunct statement for:


    The first case: .................p is false and q is false hence:


    ..............................(2) is false.


    For the next case :..............p is true and q is false hence:



    .....................................(2) is false.


    For the next case:..............p is false and q is is true hence:



    .....................................(2) is false.



    Now the converse of (1) is the conditional, q------->p....................3


    And:


    For the first case :.........again q is true and p is false ,but now :


    ....................................(3) is false.


    For the next case :............q is true and p is true hence:


    ......................................(3) is true.

    For the next case:...............q is false and p is false hence:



    .......................................(3) is true.


    The contrapositive of (1) is ~q------>~p.................................4


    But now since (1) and (4) are equivalent they have the same truth table and hence for all the above cases (4) has the same values as (1).
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