# 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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https://brainmass.com/math/discrete-math/discrete-math-proofs-56770

#### 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.