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    Discrete math proofs

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    (See attached file for full problem description with proper symbols and equations)

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    1)Prove that for any non-empty sets
    A x (B-C) = (AxB)-(AxC)

    2) Let a,b be integers and m a positive integer. Prove that:
    ab = [(a mod m ) * (b mod m) mod m ]

    3)Prove or disprove (a mod m) + (b mod m) = (a+b) mod m for all integers a and b whenever m is a positive integer.

    4) prove that
    floor(n/2) * ceiling(n/2) = floor (n2/4)

    5) For any integer n show that 7n+1 and 15n+2 are relatively prime

    6) By induction show that
    1*2*3 + 2*3*4 +...n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4
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    https://brainmass.com/math/discrete-math/discrete-math-proofs-56770

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    1)Prove that for any non-empty sets
    A x (B-C) = (AxB)-(AxC)
    Proof. By definition of Cartesian product, we know that
    For every (x,y) in A x (B-C), we have . Since

    Hence,

    2) Let a,b be integers and m a positive integer. Prove that:
    ab = [(a mod m ) * (b mod m) mod m ]
    Proof. Assume that a= p mod m and b= q mod m.
    Then we know that
    a=km+p
    b=lm+q
    for some integers k and l.

    So,
    ab=(km+p)(lm+q)=m(klm+lp+kq)+pq
    Hence,
    ab= pq mod m.
    That is
    ab = [(a mod m ) * (b mod m) mod m ]

    3)Prove or disprove (a mod m) + (b mod m) = (a+b) mod m for all integers a and b whenever m is a positive ...

    Solution Summary

    There are a series of discrete math proofs here regarding sets, relative primes, floor and ceiling, and modulo arithmetic.

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