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

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