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Homework 3 CSE 191 Fall 2016 Due: Friday 10/7/2016

 

This assignment has some examples of different proof methods as well as problems regarding set

 

notation. There are a total of 100 points on the assignment.

 

(1) (8 points) Prove that there is no positive integer n such that:

 

200 < (n + 1)2n < 300

 

Hint: try a proof by cases.

 

(2) (8 points) Prove or disprove the following statement:

 

If a and b are rational numbers then ab is a rational number.

 

(3) An integer m is a multiple of an integer a if there exists an integer k such that m = ak.

 

E.g., m is a multiple of 2 if there exists an integer c such that m = 2c.

 

E.g., m is a multiple of 5 if there exists an integer d such that m = 5d.

 

(8 points) Prove that if n2 is a multiple of 5, then n is a multiple of 5.

 

?

 

(4) (8 points)

 

Prove

 

that

 

5 is not a rational number. (hint: look at how the book/lecture

 

?

 

shows 2 is not rational)

 

(5) Let A, B, C, and D be the following sets:

 

A = {1, 2, 3, 4}

 

B = {1, 1, 3}

 

C = {1, {3}, 1, {1, 3}, {3, 3, 1}, {{1, 3}}}

 

D = {2, 2, 2, 3, 3, 4}

 

Provide the answer to the following questions about the above sets, providing a short

 

justification for each (1+1 points each, for a total of 32 points).

 

(5a) Is B ? A?

 

(5b) Is B ? C?

 

(5c) Is B ? D?

 

(5d) What is |A|?

 

(5e) What is |B|?

 

(5f) What is |C|?

 

(5g) What is |D|?

 

(5h) Is B ? A?

 

(5i) Is B ? C? (5j) What is A ? C?

 

(no justification needed for (5j))

 

(5k) What is A ? C?

 

(no justification needed for (5k))

 

(5l) What is B × D?

 

(no justification needed for (5l))

 

(5m) What is P(B)?

 

(no justification needed for (5m))

 

(5n) Is B ? P(C)?

 

(5o) Give a partition of A by 3 sets.

 

(5p) Does a partition of B by 3 sets exist? (6) For each of the following statements, state if it is always true for any two set s A and B. If

 

the statement is not always true, provide a counter example (4 points each, for a total of

 

20 points).

 

(6a) If A ? B, then A ? B.

 

(6b) If A ? B, then A ? B.

 

(6c) If A = B, then A ? B.

 

(6d) If A = B, then A ? B.

 

(6e) For any two sets, A and B, if A ? B, then A2 ? B2

 

(7) Using the set identities in section 3.5 of the zyBook, name the set identity used to justify

 

each of the identities given below (2 points each, for a total of 10 points)

 

(7a) (B ? C) ? (B ? C) = U

 

(7b) A ? (A ? B) = A

 

(7c) A ? (B ? C) = A ? (B ? C)

 

(7d) (B ? A) ? (B ? A) = (B ? A)

 

(7e) ((A ? B) ? C) ? ? = ?

 

(8) Prove or disprove that ? = {?}. (6 points)

 


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