Problem 1: Sets

Which of the following sets are equal? Show your work step by step.

• {t : t is a root of x² 6x + 8 = 0}
• {y : y is a real number in the closed interval [2, 3]}
• {4, 2, 5, 4}
• {4, 5, 7, 2} {5, 7}
• {q: q is either the number of sides of a rectangle or the number of digits in any integer between 11 and 99}

(Solution)

1

Homework #2 2

Problem 2: Cartesian Product of Sets

Explain why (A B) (C D) and A (B C) D are not the same.

(Solution)

(25 points)

Let A, B and C be sets which have different cardinalities. Let (p, q, r) be each triple of A B C where p A, q B and r C. Design an algorithm which finds all the triples that are satisfying the criteria: p q and q r. Write the pseudo code of the algorithm in your solution.

For example: Let the set A, B and C be as A = { 3, 5, 7 }, B = { 3, 6 } and C = { 4, 6, 9 }. Then the output should be : { (3, 6, 4), (3, 6, 6), (5, 6, 4), (5, 6, 6) }.

(Note: Assume that you have sets of A, B, C as an input argument.)

(Solution)

for write a condition do

end

When you want to write a while loop, you can use:

while write a condition do

If you need to return, use return end

For any additional things you have to do while writing your pseudo code, Google How to use algorithm2e in Latex?.

Homework #2 3

Problem 4: Relations (3+3+3+3+3+3+3=21 points)

Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if (a) x 6= y.

(Solution)

• xy 1.

(Solution)

• x = y + 1 or x = y 1.

(Solution)

• x is a multiple of y.

(Solution)

• x and y are both negative or both nonnegative.

(Solution)

• x y². (Solution)
• x = y².

(Solution)

Let f be the function from R to R defined by f(x) = x². Find

(a) f¹ ({ x | 0 < x < 1 })

(Solution)

(b)f¹ ({ x | x > 4 })

(Solution)