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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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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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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 ...

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  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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