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Discrete math proofs

(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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Solution Preview

Please see the attached file.

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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