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Discrete structures and logical equivalences

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Q1) Use the standard logical equivalences to simplify the expression
(ㄱp ^ q) v ㄱ(pVq)

Q2) consider the following theorem
"The square of every odd natural number is again an odd number"
What is the hypothesis of the theorem? what is the conclusion? give a direct proof of the theorem.

Q3) consider the following theorem
" The sum of a rational number and an irrational number is an irrational number.
What is the hypothesis of the theorem? what is the conclusion? Give a direct proof of the theorem

Q4) Prove that for any integer n, 3ㅣn^3+2n (Hint, consider 3 separate cases)

Q5) For the following sets A and B find A∪B, A∩B and AB.

a) A={1,2,a} B={2,3,a} b) A={2,7,b), B={7,3,4} c) A=Z, B=N

Q6) Write down the power sets for each of the following sets:

a) φ b) {φ} c) {4,7}

Q7) Find the Cartesian products A*B, B^2 and A^3 for the sets A={0,x} and B={0,1,4}.

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Solution Summary

The following posting provides an example of working with logic (theorems and statements), sets, and proofs.

Solution provided by:
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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!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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