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Math Help - descrete math

  1. #1
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    Question descrete math

    1. Use Venn diagrams or membership table to demonstrates that:
    A U (B ∩ C)= (A U B) ∩ (A U C) is a true equation for any sets A, B, and C. If you use Venn diagrams , show the three circles overlapping and include a legend pointing out which set is represented by witch shaded regions.

    2. Find the best conclusion for this Lewis Carroll problem . The "best" conclusion is one which uses all of the given information.
    A. No one who is going to a party ever fails to brush his hair.
    B. No one looks fascinating , if he is untidy.
    C. Opium -eaters have no self-command.
    D.Everyone who has brushed his hair looks fascinating.
    E.No one wears white kid gloves unless he is going to a party.
    F. A person is always untidy if he/she has no self -command.

    Assign letters to the statements , write the premises using logic symbols , write a valid argument which leads to the best conclusion , then write the conclusion in words.

    3. Prove this statement : For all X ε { 2,4,6,8,12,10}, 3x +12 is an even number.

    Please help !!! Thanks
    Last edited by olenka; October 9th 2008 at 03:31 PM.
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  2. #2
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    Question 1 and 3 are asking you to prove something false. Perhaps you have made a typo?

    Question 2

    The "best" conclusion concerns the relationship between opium eaters and white kid gloves, since these are mentioned only once. Use the other statements to get a chain of implications from one to the other.
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  3. #3
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    Question Discrete math !!!!!

    Yes, I have made a typo, but now it is corrected! Thaks

    Quote Originally Posted by badgerigar View Post
    Question 1 and 3 are asking you to prove something false. Perhaps you have made a typo?

    Question 2

    The "best" conclusion concerns the relationship between opium eaters and white kid gloves, since these are mentioned only once. Use the other statements to get a chain of implications from one to the other.
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  4. #4
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    1. Use Venn diagrams or membership table to demonstrates that:
    A U (B ∩ C)= (A U B) ∩ (A U C) is a true equation for any sets A, B, and C. If you use Venn diagrams , show the three circles overlapping and include a legend pointing out which set is represented by witch shaded regions.
    If you are doing this using Venn diagrams, draw 2 of them, one for the left hand side and one for the right. Then colour in A and (B ∩ C) on the left one and (A U B) and (A U C) on the right one. All of the coloured parts in the left diagram will be the same as the parts on the right diagram that are coloured in both times.

    3. Prove this statement : For all X ε { 2,4,6,8,12,18}, 3x +12 is an even number.
    Try each value for x in the set. If 3x+12 is even each time then the statement is proved.
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  5. #5
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    Exclamation Discrete math !!!!! help!

    Hi thanks for all hints but I am still a little bit confused about the problem #2!!!
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  6. #6
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    I think your problem is just understanding the task rather than any problem with understanding how the logic works, so I will do part of the question.
    write the premises using logic symbols
    I will do C.
    x \in opiumEaters \implies x \not \in selfControl

    write a valid argument which leads to the best conclusion
    C. Opium -eaters have no self-command.
    F. A person is always untidy if he/she has no self -command.
    Together these tell you that Opium eaters are always untidy. Look for the other statement concerning untidy and combine this with the fact that opium eaters are untidy. Continue until you have used all the statements.
    Last edited by badgerigar; October 11th 2008 at 02:47 AM. Reason: latex error
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  7. #7
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    Quote Originally Posted by olenka View Post
    1. Use Venn diagrams or membership table to demonstrates that:
    A U (B ∩ C)= (A U B) ∩ (A U C) is a true equation for any sets A, B, and C. If you use Venn diagrams , show the three circles overlapping and include a legend pointing out which set is represented by witch shaded regions.

    2. Find the best conclusion for this Lewis Carroll problem . The "best" conclusion is one which uses all of the given information.
    A. No one who is going to a party ever fails to brush his hair.
    B. No one looks fascinating , if he is untidy.
    C. Opium -eaters have no self-command.
    D.Everyone who has brushed his hair looks fascinating.
    E.No one wears white kid gloves unless he is going to a party.
    F. A person is always untidy if he/she has no self -command.

    Assign letters to the statements , write the premises using logic symbols , write a valid argument which leads to the best conclusion , then write the conclusion in words.

    3. Prove this statement : For all X ε { 2,4,6,8,12,10}, 3x +12 is an even number.

    Please help !!! Thanks

    problem 2 can be tackled in two ways,either by appealing to symbolic logic or by using ones very strong thinking without using symbolic logic.

    if we want to use symbolic logic we must convert the above sentences so that we will be able to use letters.

    hence:

    A CAN be converted to:if x is going to a party x brush his hair.

    B can be converted to: if x is untidy then x does not look fascinating.

    C FOR c we have :.....if x is opium eater then x has no self command.

    D for D we have:........if x brushes his hair then x looks fascinating

    E for E we have:.........if x goes to a party then x wears white kid gloves

    F for F we have:.........if x has no self command then x is always untidy


    Now let , x is going to a party=p.......x brush his hair=q..........x is untidy=r..........x does not look fascinating=s.........x is opium eater=t............x has no self command=u.............x wears white kid gloves=v .

    And the above now is converted into symbols as follows:

    A....................................p------->q

    B.....................................r-------->s

    C....................................t-------->u

    D....................................q-------->~s

    E....................................P--------->v

    F....................................u--------->r

    now you can use the laws of propositional calculus to get the desired result.

    The result is if i am not mistaken is p------->~t or v

    FOR example from A and D and a law which is called hypothetical syllogism you get p----->~s............................................... ..............1

    From B BY using contrapositive law you get ~s----->~r.......................................2

    FROM 1 and 2 and using again hypothetical syllogism you get: p------>~r............................................... .......3

    From F and contrapositive you get : ~r------>~u............................................... ......4

    From 3 and 4 and using again hypothetical syllogism we get------->~u............................................... ........5

    So far we have used A,D,B,F AND we left with C and F

    Don't forget we must use all hypothesis

    From C and using contrapositive we get: ~u-------->~t............................................. .6


    From 5 and 6 and using hypothetical syllogism we get------>~t............................................... ..7.

    you do the last step yourself
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